First option:
Pros: I get a checked bag and I know I like the building. Even though I'm using miles I think the price is decent.
Cons: You can only select "one bedroom" and not pick out the exact apartment you want. They're all individually owned so some have different layouts, etc. Some have tubs, others just walk in showers. Some have Nespresso machines (I love mine, lol). It's also just a 1 br instead of 2.
I'd say no. You'd be expecting the same great place you stayed last time and might get somewhere much worse. I think it would be hard to be in the building you like, knowing the "perfect" apartment is right there and not being able to stay in it. And did I understand this correctly? $1K in taxes to use your miles? I've never had a card that offered miles, so I don't know how that would work.
Option 2:
Pros: Able to pick the exact apartment I want.
Cons: Can't use miles.
Ummm...maybe?
Option 3:
Pros: I'd have the place I really liked.
Cons: Even if I was able to get them to agree to the discount, it's still on the pricier side. But a great place to stay.
I think what it comes down to IMO is how important is it to have the best place to stay? If you're planning on just going, going, going most of the time you're there, and the place you return to at night is simply a place to sleep, I'd concentrate more on finding somewhere in a good location that won't be too expensive. Sounds like you've ruled out all hotels...even budget stays? I've never traveled abroad, so I don't know if hotels outside of the US offer amenities like complimentary breakfasts or fridges in the room or whatever else we spoiled Americans demand <g>.
Whatever you decide....eeee, London! Sounds like an amazing trip! And thank you for letting me participate in a RL version of House Hunters <g>.