Probably not. It doesn't matter which browser you use to surf the web as the trainer will use IE's technology to do it's authentication process (this isn't changeable).
Does 'probably not' mean 'I'm not sure'? So why doesn't the existing trainer(s) work? I have IE10 installed and up to date as far as Windows Update is concerned. As old as Alice: Madness Returns is I wonder if EA is using it to test out a new anti-cheat scheme?
[Edited by Ratsneve, 4/29/2013 8:36:10 PM]
Without access to what your environment setup is like, logs, etc, I cannot determine this at all (logs detailed to even the most mundane of things are absolutely invaluable for troubleshooting, without this it's like going to the doctor and telling him you have pain and offer no further details). Seeing that others have gotten the trainer to work successfully all we can tell you is to examine your environment to find out what's causing it to be blocked.