Would fastmail (or the likes) be an ok service to use for a low volume Saas to send the regular stuff, like creating an account, resetting a password and such?
Depends on how low a volume. At my workplace we were actually using Rackspace for automated emails, but we ended up constantly blacklisted. Customer support directed us to use Mailgun for those types of emails. They said automated emails required a different setup to maintain their reputation and not look like spam. I imagine it's the same case for Fastmail.
You can check out their limits here: https://www.fastmail.com/help/account/limits.html
For the $50/year account they have a daily sending limit of 8,000 emails a day.