I'm in a similar situation. My upcoming flights to Scotland were 25% less per person less with Aer Lingus than anyone else, so that was a no-brainer. My 2 travel companions won't be able to meet the carryon limit, but I've decided to try "competitive light packing" (competing with myself) and it's honestly been pretty fun! (Note, I'm not doing this JUST for fun. My own checked bag allotment is going to my daughter's bag that we'll bring back with us. She's a uni student in Scotland, but will be coming home for the summer. She has her own travel planned after our trip, and won't be able to store her larger bag during her own trip, so we'll bring it home with us.)
The bag I bought is a Deuter Aviant 28L with a suitcase style opening and cinch straps, plus a roomy business pocket for laptop and office needs (not needed for this trip, but likely useful in the future). The Aviant is a bit too long, but the back is a stiff foam that can bend a bit. I'm confident I will be able to push it into a sizer. There just aren't that many backpacks less than 19" unless you get a "school" backpack, and the shape of those is inefficient for maximizing your volume allowance. The Aviant bag is over 2 lb., but that's still amazing compared to my 5 lb. 21" rolling backpack or my previous 7 lb 22" spinner. It's got some padding, and therefore some wasted weight. On the suitcase side, I'll have to put everything under the cinch straps, and leave the large "lid" compartment empty in order to meet that 8" depth. But it can be done!
Another bag I saw that was intriguing was the Rangeland Business backpack I saw discussed on this forum. That one is even smaller, but very boxy and effiecient, and works for Lufthansa.
I have had to weigh things... yes, I am becoming one of those. It's been fun building a spreadsheet and comparing options. I picked up a few lighter-weight substitutes for things I already own, and I've been sorting through my packing supplies to make sure I'm making good choices. One of my favorite new items is a miniscule sil-nylon toiletry bag from Sea-To-Summit. It actually holds everything I need and I saved a full half-pound with this bag! I'm currently struggling to find a unicorn bag - something large enough to be my airport bag but also small enough to look like a "purse" for daily touring/hiking, nylon, well under 1 lb., and if I can find it, convertible from crossbody to backpack. I don't want a shapeless sack and I don't want a massive tote bag. If I manage to find this elusive wonder, I won't bring a separate purse/camera bag and a packable backpack. My existing bags for those two items are a full 24 oz., and I just know I can do better.
Another area to watch your weight is electronics. My tablet and teeny folding keyboard are 1.5 lb, replacing a 3 lb. laptop. Someday I may ditch the tiny tablet and just use my phone, but the larger screen is so nice for so many things. Cords and charging blocks add up - consider a single charging block with multiple ports - it should have both USB-A and USB-C if you have any newer electronics. Think about how many cords you really need - do you charge everything every day? Could you use one cord for multiple devices? At 1 ounce each, cords add up.
Universal adapters are great, but very heavy; perhaps a dedicated adapter would be better.
And of course, toiletries - use small (small!) containers, bring only what you need, be willing to use a few hotel toiletries, and accept that you may be visiting a pharmacy to fill in if you miscalculated.
I'm currently 1 lb. under weight in my carryon, though my personal item will be heavy with my camera and tablet. But I can't put either of those in an overhead bin; they have to stay on my shoulder. Happily, I find I don't have to make any compromises. But I also won't be buying many (any?) souvenirs.
But I would probably check my rolling backpack if I could. Good luck!