This is what I know about the British Museum in September (haven't been there in April): A museum security person told me weekends and Fridays are busiest, plus rainy days. If Friday works best for your itinerary (or if you want to take advantage of the late hours), you can't avoid the Friday situation, but you might want to look up at the sky and consider the weather forecast. If it starts out rainy, the morning will be busier than usual. If the day starts out nice and then clouds roll in, additional mobs will descend on the museum.
I am a late starter and have never managed to get to the Museum before opening time. Arriving shortly after opening time means about 30 minutes in line, on average. The security line may well be shorter an hour or two later (unless it has just started to rain!), but I can't say that for sure. There's a second entry point on the back side of the museum used by tours, but others can normally enter there as well (not if the scanners are broken, which happened on the day of one of my visits). That line is usually shorter, but it's a rather long walk around the building from the front, and if you arrive and discover there's a problem, as I did, you'll have to walk back around the building and will have wasted a lot of time.
You're right to think about the food situation, because I don't think anything was open after about 4 PM on the Friday I went to the museum, and everything except the grab-and-go place on the ground floor may have closed by 3 PM or so. There are quite a few cafes/restaurants south of the museum, but I've never eaten in any of them, and going to one of those mid-visit would mean facing the security line again. Of course, the security line might be extremely short at 5 PM or so; I don't know about that.
I made multiple trips to the museum in 2022. The last two were to the upper floor, which is always noticeably less busy. On my Friday visit, that floor was blessedly empty by late afternoon/early evening. I don't know what it was like on the ground floor, but I'm sure it was better then than earlier in the day. People do get hungry.
If you decide to go later in the day so you can take advantage of less crowding at the end of the day, I'd suggest going upstairs first and finishing on the ground floor. That will be tricky from the time-allocation standpoint, because I suspect you'll want to spend more time on the ground floor, where the kids may be more wowed by things like mummies. It might be helpful to think about what would be the maximum time your group could stand be on its feet in a museum. Most people have a limit, be it 2 hours or 4 hours. A sit-down meal along the way could extend that, of course. Three hours seems reasonable to me.
