I assume you will have to run the daemon and the wallet service then point myTrezor.com to the wallet service as the backend instead of Bits of Proof. Uncharted waters here for me sounds like fun.
That's where the proposition loses me (using a web browser), not to mention that I have a feeling that Bitcore might be a somewhat of a can of worms..... I wouldn't really trust Bitpay (who develop Bitcore) to compile their own binaries, so I would have to do that for myself. And what if their tagged source won't compile? Contact Bitpay for help? There goes some of your anonymity straight away. Don't like it.