A few thoughts...I'll say up front that I do agree with the above about generally having your trip all planned before you pull the trigger on anything, but what's done is done and we all learn somehow, so we'll work with what we have:
- If you want specific help, provide specific dates. No one is going to steal your identity or your soul or whatever based on knowing you are from MKE and going to Europe for 8 specific days in the summer. If you have specific dates we could help down to providing advice at the property level.
- That being said, picking general dates in August (I used Tues-Sun 8-13 Aug so it at least covered a weekend) and searching 2 adults + 1 child, there are some options out there. I'm seeing 4 or 5 Doubletree and Hilton brand properties with 2-double, double + single, queen + sofabed setups in the ~50-70K point/night range (so 200-280K for 5 nights using 5th night free). Are they the absolute nicest places in town? No, but perfectly serviceable.
- I'll note that it's that 3rd person that is the killer - if you search for just 2 adults there are a ton of London options at reasonable prices. Because adding the 3rd person gets you into the territory of multiple larger (e.g. not twin) beds, or bed + sofabed, that becomes the domain of "premium" room types, and thus astronomical points redemptions. It may be helpful to search 2 adults and then scour the results to see if any workable room types get mixed in - perhaps something that was miscoded but could actually sleep 3 (but be darn sure!).
- Do you have the opportunity to perhaps go out of chain for other currencies? If you and your spouse don't have MR cards, you probably have time to open a couple and earn some MR bonuses. What about Chase points or such - transferrable to Hyatt, MR, etc.?
(Editorial note: How on earth do I get a blank line before a bulleted list? No matter how many I add in the editor, they all disappear when I post...I don't think it used to be this way)