Five months is not short notice, full stop. Even three months I would have a hard time thinking of any reasonable argument. The length of time you've used a service is irrelevant, the question is how much of a lift is it to switch, and nothing on Google Apps would take you anywhere near 90 days to switch out of. I pay something like $65 or $70/yr for O365 Outlook for a personal domain email, and the email is the only thing I get with that, so this is on the cheaper end to boot. I'm sure you can find custom email domain handling for less though.
> What do you think the perception will be if Google asked them to pony up $60 a year [for Gmail] with 5 months’ notice?
Realistically, I and a lot of other people would think "remind me in 4 months and ~25 days so I can put my credit card in." But it's also not a valid comparison because in Gmail you're the product via advertising. Not the case for Apps.