I cannot directly install modules from CPAN because the compilers are turned off to shared hosting users.
You can directly install Pure-Perl modules from CPAN as those don't require any separate compilers.
If you are really stuck and need an XS module you could always cross-compile as a last resort. However, in your shoes I would break the shackles at this point and move off shared hosting and onto a VM. Your shared hosting will probably limit the scope of any persistent framework/middleware anyway.
I have gone through this list
That list contains things which aren't web frameworks and is pretty far from exhaustive. Earlier this year I submitted this quest response which is far more extensive (and even then I left quite a few out).