Fastmail is amazing, yes.
I was even more impressed when I ran into some messages from one of its principals on a linux kernel mailing list (he found a real bug in the VM system).
And I was impressed even more when I submitted a bug report with Fastmail support (about a particular IMAP command) and got a response from him, a very unusual experience.
They are working hard on expanding their business, and I sincerely wish them all the best.
However, back to topic:
As far as I know, Gmail does not have a per-IP limit -- it's truly per account, and is enforced at login time.
There is a good chance that a desktop mail app is just too "greedy" about server connections -- enabling push for every folder it finds in the account (and since Gmail has quite a few predefined folder... well...)