RE: gdbserver with a core file?

I don't know if anyone has added such functionality to gdbserver; but, with regards to
the problem you're trying to solve -- I've faced similar ones in the past.

If you use emacs, you might consider gud plus a small shell script or two.

I've used in in the past to solve the problem where I have GDB and the executable
remote but sources local.

In emacs, do

M-x gdb

when prompted for how to run gdb, replace gdb with the path to a shell script that

. sets up the local environment (if needed)
. invokes ssh -- something like:

    ssh remote-host /path/to/remote/shell/script

where the remote shell script does a cd if needed, creates a gdb init file if appropriate
and / or needed, and then invokes gdb with appropriate arguments.

GDB communicates with Emacs.  When you stop / step / next / whatever,
GDB tells Emacs what it needs to know.  Sources stay local.