I think this is a good suggestion, and it sounds like you are quite analytical so might also enjoy it. The sites you mentioned do not lend themselves to one recommendation (I can see why you have been spending a while considering where to stay BTW - there are loads of Hilton properties in London and the choice can feel completely overwhelming). Take the museums. The British Museum is the #1 visited but it is in Bloomsbury and nowhere near the main "museums" area which is in South Kensington.

I think this sort of process might help with your decision. You have a very good budget so that should be fine. If you want everyone in one room then there are various threads here which have discussed London options. However your budget may allow for a suite. If you are really focused on a good executive lounge then (based on my understanding of current reviews, not having stayed at any of these recently) Bankside, Tower Bridge or the Conrad. Of these three Conrad would best fit your overall criteria. It is close to St James Park tube (which is on the Circle/District linking with no changes to both South Ken and Tower Bridge), very short walk to the Abbey, close to the Palace. I would say the whole area there is quite easy to walk around (but there are a lot of tourists!). Bankside is not particularly close to the tube, and not near anywhere you list, and Tower Bridge is somewhat far east for most of your plans.

I wonder about Park Lane. The South Ken museums and Palace are close. It is near the entrance to the Piccadilly Line tube station which is a very good line for tourism, and there are a number of bus routes which would fill in a few of the gaps too. If you can afford a higher cat room the views are fabulous.