I suggest that you also monitor the 3rd and 4th night free options when doing FHR on the hotel website directly you are considering. Sometimes the hotels advertise the free night but it is not offered on the FHR site or Amex doesn;t see it when speaking to an agent. The free nights in my experience are almost always combinable if you talk to AMEX and ask them to arrange it with the hotel. Also, some of the hotels will also guarantee early checkin (Berkeley, Peninsula and Lanesborough currently offer on their website.) This is almost always combinable as well if you insist).
I struggled with a similar choices but landed for some of the same reasons mentioned on the Lanesborough. Some of the room sizes however are smaller than elsewhere however and some of the bathrooms don’t have a tub and a bath (either / or). The feedback from the others in this thread was incredibly helpful when considering my choice and there is also the house cat Lilibet that sealed the deal for us! If booking the Corinthia or Cafe Royal, GHA Discovery Platinum status can be had by buying a subscription to Business Traveller Magazine around $30 USD. (Get you some limited GHA Perks). FoundersCard has interesting rates at the MO Hyde Park but not for all room categories. Bon voyage!