I've never flown AT so can't speak about that one, but if you have AY or QR on an xONEx, it would be difficult to avoid since both can be sticklers for checking carry on weight, especially when not checking anything in. This is where I love airports that have cheap storage lockers like in Japan, so if you know they are only going to weigh carry on's at the check in counter and not the gate you can easily hide the heavier stuff until after it's been checked. I've had to do that with Peach and Air Asia before.

But on the OP's itinerary, both AA and BA will allow you to carry on the kitchen sink per-se so would be less problematic as long as the bag was small enough.

I'm often gone for 3-4 weeks on my xONEx's so packing into a carry on compliant bag isn't feasible (often dealing with 2 climate extremes as well), I always check in one (or two bags after shopping along the way), haven't had a problem yet on them, but none included LHR either. I did just have one of our three bags get delayed a day in LHR on a different BA itinerary last week. Took them 4 days to finally get it to my home but BA is pretty generous on what constitutes "essential items" that need purchased so in the end it wasn't so bad.