If your mail is starting from a Bigfoot address, send it direct from there to your Gmail Imap address. You don't need to "use" the BTInternet address at all.
You can also set the BTInternet one to point at Gmail.
I don't use the e-mail addresses provided by my ISP. I've changed ISP a few times over the years, and not had to tell anyone about it. It "just works".
Once you've had a few weeks working with Imap on the iPhone and the home PC, you won't want to change back.