Get a bigger/stronger board than that, fiberglass (FR4) is best. Buy a bunch of breakaway .100" female headers
3M|929974-01-36-RK|PC Board Connector | Newark.com and use those on the proto PCB.
Next use some Samtec board standoff female headers that have extra long tails, to put on TQFP board
**broken link removed** and
https://www.samtec.com/ftppub/cpdf/SSQ-MKT.PDF The extra long tails can then go thru the TQFP board, AND also have the length to properly mate with the female headers that have been placed on the proto PCB.
Then buy a JE10 wire jumper kit.
Good luck.