How to interface with SMD pads to a custom MCU on PCB
The thing you're looking for is called an interposer. It's a small PCB that adapts one footprint to another.

A number of companies specialize in them. Some links below...

https://www.advanced.com/products/package-conversion-adapters/qfp-adapters

http://www.isipkg.com/wp-content/uploads/2016/09/ISI-Obsolescence-Solutions.pdf

http://www.proto-advantage.com/store/interposer.php

That said, it may be cheaper to lay out the entire board for the micro of your choice.

Another approach is to use a DIP-based assembly with your device (e.g., Arduino Nano), and connect that to the QFP using magnet wire. It's tedious, but it will get the job done. Once you've stabilized your design then you can re-layout the board with confidence.