Hi Joshua,
Based on the information you have provided so far, I see no reason why you need DNS Private Resolver. What you can do is add an entry to your local hosts file with the private IP address of the database in Azure.
If you have many hosts up in Azure (besides the database) that you want to connect to by name over the VPN, then you may want to consider using DNS Private Resolver, or a small VM running DNS that will perform same function [as DNS Private Resolver].
If you are unsure how to make entry to your local hosts file please let me know and I will explain further.
Please click Accept Answer and upvote if the above was helpful.
Thanks.
-TP