Emacs inherits its environment from the parent process. How are you invoking Emacs - from the command line, or some other way?
What happens if you:
M-x compile RET C-a C-k bash -i -c your_alias RET
Invoking bash as an interactive shell (-i option) should read your .bashrc aliases.
Edit: I think both M-x shell-command and M-x compile execute commands in an inferior shell via call-process. Try the following in your .emacs (or just evaluate):
(setq shell-file-name "bash")
(setq shell-command-switch "-ic")
I notice that after evaluation of the above, .bashrc aliases are picked up for use by both M-x shell-command and M-x compile, i.e
M-x compile RET your_alias RET
should then work.
My environment: Emacs 24.1 (pretest rc1), OSX 10.7.3