Trouble finding a personal item due to size restrictions

Well, you always take a risk when you circumvent the rules, but they're usually stricter with the carry-on bag, rather than the personal bag.

A lot depends a bit on the bag itself. If it's soft-sided and flexible, your odds of getting it on board are much greater than if it is stiffer and hard-sided. Just don't overpack it—in fact, if you can pack less in it, and flex it up a bit so it looks smaller, so much the better. And of course, you have to be able to fit it under the seat.

Have you looked at the Rick Steves store on this website?

One of his new products is the Oslo Backpack That measure 15.5 x 11.25 x 6. That's pretty close.

The problem with short length backpacks is that they tend to ride high on the back - above the lumbar area. That makes them uncomfortable. The Rick Steves civita is small, but the straps/shape allow it to be carried in a “sagging” lower position which makes it comfortable. Look at tote packs, purses, zip-top duffels with over shoulder straps. Tote packs with thinner, less rigid straps that can be adjusted like the civita may be the way to go. Both Patagonia and Osprey have tote packs. I have an old version patagonia tote pack that works well for this.

Patagonia.com has the atom and the terravia. I have an old version of the terravia.

Is your backpack so stiff and/or full that it can't be made to fit into the sizer?
As far as a new bag, there have been recent posts about tote packs. Review those and see if any of the recommended ones fits the size requirements.

I use a briefcase as my personal item. Just returned from my 3rd trip with it and I’m still very pleased with using it. It is 14.2 x 10.2 x 4” and weighs 12ounces. It sort of stands up under the seat so it doesn’t flop over taking lots of foot space. This trip I was able to stand it up against my seat and it was completely out of the way. It comes with a crossbody shoulder strap and is comfortable to carry. Mine is made by Tom Bihn and is ridiculously expensive in my opinion, over $100 as I remember, but you could probably find a cheaper briefcase shape bag on Amazon if you want to think of something other than a backpack.

I am also flying BA in a few months, so I checked again, but see a different set of measurements?
" Hand bag
Up to 40 x 30 x 15cm (16 x 12 x 6in)
Up to 23kg (51lb)
Essentials like medication, mobile phones, other electronic devices and passports
To be placed under the seat
Guaranteed in the cabin"
I chose them over Lufthansa for this trip, in part because of their generous weight allowance for both carry on and personal item. Is your bag rigid or can it squish into the sizer?
I do like my Portland Gear Cascade (smaller version) backpack and it's a shorter height.
Hope this helps!

IME the dimension that crushes one's hopes is not the height or the width, but the depth (thickness).

I use one of these and it has been exactly what I wanted: HOMIEE Ryanair Personal Item 40x20x25 Travel Bag

It's big enough to be useful (and when fully expanded, holds a ton of stuff), it's surprisingly roomy and functional, but can be cinched down (squeezed) to meet even the smallest sizes that may be necessary. It's light weight, well-made, nicely padded and feels great on my back even when full of camera gear - and is gob-smackingly inexpensive (30 bucks last time I checked).

Dimensions are 40 cm high x 25 cm wide x 20 cm deep, or 15.75" high x 9.84" wide x 7.87" deep - so technically it's probably about 1.75" too thick to meet BA's stated limits - but here's the kicker: it has external compression straps that allow you to flatten it down to just 4" to 6" deep, allowing it to pass muster from even the stingiest airlines. It's soft-sided (no hard, unyielding plastic, every dimension is squishable to some degree); the external compression straps can get it down really thin...as thin as you need it to be (or if they're not being too strict, it'll hold a surprising amount of stuff). It's also totally rectangular, with nice, square, 90-degree corners - so no space is wasted (a pet peeve of mine).

Now, to be fair, when you cinch it down all the way with the compression straps, it's pretty much a pancake, so you are not going to fit a whole lot into that thin profile. But what I love is the flexibility to make it as thick or as thin as needed for that day's flight. If you really need to get it down to just 5" thin (and you might), you just need to pack it accordingly - or you need a Plan B (which I call....Cargo Pants!).

Remember, you never know exactly how strict they're going to be - but you need to be prepared to do what you have to do to get on that plane, even if the gate dragons have just had a fight with their spouse and are in an ugly mood. My strategy, using this as my "small personal item" is to pack it somewhere between "half-full" and "everything including-the-kitchen-sink". If they're being very strict, I remove just as much of the contents as I need to, transfer those to my pockets, cinch it down thin enough to get through, and I'm good to go (so when you pack it, consider which items you are putting in there - some may wind up in your pockets until you're on the plane, which may be a minor inconvenience but is not the worst thing that can happen). If they are not being 100% strict and checking carefully (which is sometimes the case), I smile and walk on board with everything in the pack. If you can't abide the idea of stuffing your pockets at the last minute to board, then just pack it partially full, cinch the straps down as much as you need to, and off you go. Both strategies work for me.

I love this little backpack, it has become a favorite. I was extremely skeptical when I bought it (under 30 bucks for a pack?) but it has been great. I'm planning to take it aboard several BA flights this summer and I'm confident it's going to work for me. We'll see about that and YMMV/no guarantees...but I'm a believer (who wears cargo pants when necessary...).

Replying to several people, thanks everyone for your input.

The backpack I planned to use can't be squished down in height. Really none of the dimensions are small enough, so I think it's out.

I think the measurements of 16x12x6 are essentially the same as what I got. I used a converter to change from cm to inches, and did not round up.`Most of what I've found that comes close to fitting is 16x12x6, and there aren't many. But there are a few. Whoever said depth is the problem, you are right!

Regarding the Ryanair backpack, I actually put that in my Amazon cart earlier. Is it comfortable to wear? Bearing in mind the comment upthread about how they sit on the back. I'm short, so maybe it won't matter as much? I actually have never used backpacks.

Would they allow a belt bag worn across the chest in addition to the carry-on and personal item? Or is that one too many? That might give me a little more flexibility.

Thanks again, everyone who has commented.
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Would they allow a belt bag worn across the chest in addition to the carry-on and personal item? Or is that one too many? That might give me a little more flexibility.

It depends on the gate agent. Some might allow it, some might not. I guess it depends on the size and what kind of mood the gate agent is in. But, technically, it's a third bag.

I'm guessing, and this is only a guess, they are more concerned with the main carry on. I've flown BA many times and have never seen the personal item checked as long as it wasn't massive.

But that doesn't mean they won't.

Of course, the question you could ask yourself is how much stuff do you have to take in your personal item?

Oh ... the comments are just starting. It’s Sunday morning!
I strongly suggest that you look at the patagonia.com terravia tote pack. It is structureless and completely squishable in all dimensions. Lightweight. Straps can be adjusted. Water resistant. Durable. 24 liters of space for packing. Some basic, functional pockets. A bit expensive for what it is. However, Patagonia uses responsible business practices.

Negatives: not good for computers. Not good for 6 packs of pop or carrying wine bottles. It can carry 6 packs and wine bottles, but not comfortably. Use packing cubes in the small, medium size.

My observation has been that there is very little enforcement of bag size or number. I check my bag so I do notice others as they line up. Two carry on sizes, huge packs and a carryon roller. Three bags. All not uncommon. From your size bag I would say you would have no problem whatsoever.

I have two personal item backpacks that just about meet BA's requirements. A Fjalraven 15" laptop bag at 15.75 x. 10 x 6.5 and a Babolat tennis backpack at 15.5 x 11.5 x 7. Both are designed for adults. Both are squishable. Both have padded shoulder straps. So if you don't pack them out to their absolute limit you shouldn't have a problem. I have not flown BA with them. But have flown Condor, Lufthansa, Aer Lingus, American and Alaska without an issue.