It doesn't sound practical for all the reasons listed OP, but I don't understand the amount of judgment about kids getting tired walking on here! I've noticed it before. It's like there are some mumsnetters that don't understand there is grey area between being a baby who can't walk at all and a fully grown adult with long legs who can walk many miles! And think this makes them morally superior.
I think 4/5 were awkward ages with our kids for days with lots of walking. 'Too old' to use a buggy, too big to carry, but still tiny legs and not quite mature enough to just 'push through' tiredness and control their emotions. I often think they are still slightly prone to regressing to toddler at that age when tired / upset / overexcited etc. We've just come back from a couple of days in London with my 7 year old and it was noticeably easier than when we took his sister at 5 in terms of stamina / mood!
A day out at a big museum and using public transport to get there does involve more walking than a typical day for most people, and 4 and 5 year olds have much smaller legs, so of course they get tired! We did London museums recently and we all agreed that kind of slow shuffling around on our feet all day, on hard floors, was somehow more tiring on our feet than just going for a brisk walk in the countryside. And no it's not the end of the world if they get tired, but it's supposed to be a fun day out not an endurance test so you are not being unreasonable for wanting to mitigate that a bit!
I remember we took the pushchair to Legoland when DD was 4, the last time we ever used it for her I think, because we only had one expensive day there so we wanted to get round it all quickly and give her a rest when needed to cut down on queue complaining! I don't remember any odd looks. She did plenty of walking on other days.
With two though, I think you might just have to do as others have said.. build in sitting down regularly for snacks / shows and find the lowest walking routes there.