Not exactly sure I'm understanding your issue. Are you only wanting to access a certain e-mail via the web or would you like to have the web client's e-mails go automatically to your BB without the need for checking?
If you really want to use the web, I'd agree with using Opera. It's the only browser that works for me due to the login/password issues I face with the BB browser and Bolt.
If you want to forget the need for web access, you should be able to get all e-mails going straight to the BB by adding the email account (gmail, yahoo, etc) either at your BIS login page or by using the 'email settings' icon on your BB. Login using your BB account credentials, then add an e-mail account.