Yes, I do not know the implementation, but I assume GQ works without autodiff for efficiency reasons.
The simplest/best solution (in my opinion) would be to be able to pass a csv file of parameters to the fixed_param run. Currently I do not have the time, but I might fiddle with the C++ at some later time…
Edit: Backlink to my reason https://github.com/stan-dev/design-docs/pull/37#issuecomment-818523886