Size of day pack for European tourist attractions

If possible, I would move to a Crossbody bag you’re going to have problems with this backpack idea and a lot of places museums van even let you carry them a lot of times. Not to mention that they’re much easier for thieves to get into.

Trim it down leave the selfie stick behind. So many places won’t even let you use them that it’s not worth. Since it’s London, you might need the raincoat.

I cannot say if the rules are enforced at the Tower of London.
Anyway, for the items you list, if you don't want to buy anything expensive, you can probably get by with a cheap drawstring backpack. That's what I typically use as a daypack. It will definitely fit 2 rain jackets, 2 bottles and a few snacks and knick-knacks.
Downside: they are not very comfortable due to the thin straps and uneven surface once full. But you can sling them on one shoulder if you keep them light.

Where are you seeing a size stated?
I'm looking at the Tower of London website and it just says, as far as I can see, 'large bags' with no definition of what 'large' is.
Those dimensions must be metric dimensions, converted into inches, of 400 x 200 x 250mm.

Given the numbers who go to the Tower I find it incredibly hard to imagine jobsworths at the front entrance with tape measures to enforce such a petty rule.

I think everyone knows what "large bags" means.
In terms of cubic capacity your size fits within "their" size.

I've just measured my day pack which turns out to be bigger than yours. I've never had any difficulty in this country getting into any kind of museum or attraction.

If they stop you, just ditch the water bottles, and put on the jackets. Then the pack is within limits. No comment on the selfie stick.

https://www.towerbridge.org.uk/your-visit/safety-and-security

I'm seeing that bag size limitation for Tower Bridge but nowhere on the Tower of London website:

https://www.towerbridge.org.uk/your-visit/safety-and-security

But it's possible that you're also doing that one on the same day? If so, forget about the pack. The forecast will tell you if you'll need the jackets before you leave your accommodation, and if it looks like rain, wear them or tie them around your middle until needed. Put the bottles in the jacket pockets. I'd also leave the selfie stick back at the hotel as you're not allowed to use it inside any of the Tower's buildings anyway (other attractions, such as Westminster Abbey, forbid bringing one in at all, and there's no bag check there either.)

Your young daughter might have an undersized bag? My "purse" is a mini backpack that I carry on one shoulder; can't imagine they'd find it too large.

https://www.amazon.com/JanSport-Half-Pint-Backpack-Black/dp/B000E3GLWO/ref=asc_df_B000E3GLWO/?tag=hyprod-20&linkCode=df0&hvadid=198071104488&hvpos=&hvnetw=g&hvrand=11650361145054419243&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9019572&hvtargid=pla-320470572665&th=1

I think as long as it looks like a day bag and isn’t ‘luggage’ you’ll be absolutely fine. I imagine the rule is just to prevent people bringing huge backpacking size bags in. I always use a backpack and have never had any issues getting in anywhere.

At age 11, your daughter can carry her own day bag or crossbody bag.
That way you both will be able to have smaller, lighter ones.
You don’t need water bottles in London, and there are many very lightweight rain jackets.
Just be sure to keep the irreplaceable things , for both of you ,like passports, bank cards and keys in YOUR moneybelt under your clothes.
Nothing of value in the backpacks/day bags.

My crossbody was inspected briefly at the National Gallery and at Westminster, they are just looking for obvious issues. Which isn't great news if you stop and think about it ...

Have you watched any of Rick's videos? He loves his day pack, but he ensures it isn't fully packed, and often slips it under his arm when going into museums, as a lot do not allow, or frown, on back packs. You know, the whole annoying bumping into others (or priceless objects) thing in crowds.
I agree with others that you would be doing your daughter a favour by teaching her good travel skills - and letting her be responsible for her own water bottle and rain jacket - in the bag she carries.
These prints may be too much to consider, but I have it in a grey tweed fabric https://www.luglife.com/collections/convertibles/products/canter-2?variant=39615782355017 Also a bit pricey, but they are very light weight, holds a good amount and well balanced for comfort even when full. Converts from back pack to cross body bag, or hand carry. There are smaller options for your daughter. I love mine.

Thank you all for your kind advice! I'm so glad that I joined this forum and feel very welcome.
I looked at this Pacsafe sling at REI this morning and think I can get the job done with it. : https://pacsafe.com/collections/slingpacks/products/metrosafe-x-anti-theft-urban-sling

I'll skip the selfie stick and just keep the essentials. If I do use a water bottle, it can hang on the outside. I made need to buy a shoulder strap cushion.

I'm sure you can find a solution that does not cost $100, but that bag sure does the trick!