Minutes make a a difference... if you are say on AA80 to LHR (arr 2240) and BA 2644 to MLA (dep 0615) you could arrive at LGW as early as 0030 and check out of the hotel at 0500. That means 4 hours’ sleep.
You should be able to find the smallest room for around $100 at bloc or yotel. Toddler sleeps between the 2 of you on a king size bed (we’ve done it countless times).
If you think of a comparable situation, people will pay $1000 a head each way to fly say, BOS-LHR in Club rather than economy, for around 4 hours’ net sleep. They arrive in LHR much more refreshed.
Do the hotel, it’s a no brainer. And prepare yourself mentally for only 2 hours’ sleep because of delays etc, and when these delays don’t materialise you will be glad to have had 4 hours’ horizontal rest.