If you are using the domain only for the mail server, it doesn’t really matter if the DNS dies together with the mail server since the mail server is not reachable anyway.
Mails usually try to resend (i think up to 3 days) if the mail server is not reachable, other wise it will drop the mail, and probably, if the sender’s mail server is properly configured, will send a alert to the sender.
I’ve created a post on how to create secondary server for free, which can ensure your DNS is up (e.g. you use your domain to host a website elsewhere), but like I said, if mail server is down, the mail is still not reachable.
Unless your registrar’s dns can server can act as a Slave DNS, you will need to manually copy all DNS entries over. My post below creates a DNS slave which automatically copy entries from your MIAB.
You need to ensure your server is up, by ensuring you don’t install unnecessary modifications, don’t install other applications in the same server which may cause other services to be down, have enough space, have enough memory etc.