Your difficulty is with the startup time for MCR, which is still going to be a problem if you convert to .mex (mex are called by MATLAB or MCR so the runtime needs to load.)
To convert to something that does not use the runtime, you need to use MATLAB Coder. If the MATLAB features you need are not supported by MATLAB Coder, then you can attempt to rewrite your code into the supported subset.
In one of your other Questions, I suggested you could use MATLAB Automation, and someone else suggested that you could use MATLAB Production Server. Both of those would avoid having to start MCR each time.