If you do not mind some soldering,
this is an inexpensive approach:
long 40-pin stachable headers: 20 for $18
6x8cm plated souble-sided proto board 20 for $14
Works out to $1.60 per hat.
Is there a premade solderable HAT-like board
Yes, tons of these are on sale. A quick search using words like "prototype" and "DIY" revealed there's a board called "ModMyPi", and there are certainly others:
I find that strip-board or perf-board will fill the gap between bread-board and a PCB (although PCBs can be relatively cheap using pooled services).
Some stripboards will have a special area on one end where you can attach a suitable dual row GPIO header (note that if you're not stacking them then you may not need a full size header, only one big enough to ...