We ended up going with Mandrill which does offer the option to not log sensitive data ^1. Whether they log it somewhere else for the compliance reasons that Mailgun mentioned isn't mentioned anywhere in their docs or privacy policy, but doesn't seem to be accessible from everything I could find. You should never log or allow others to log password reset urls or other sensitive details.
1: See documentation here: https://mandrillapp.com/api/docs/messages.JSON.html#method-s... and search view_content_link