Is a MAC a valid way to ensure tamper-resistance?
First, you have to check if the HMAC integrity check complies to the governmental standard. In the USA, for instance, you have to check the FIPS specifications.
If that has been clarified then here are my answers:   


  is this a valid use case for a MAC?


Yes.    


  Are there any major or minor flaws with this technique?


The one you already stated.


  Is it OK to reuse the same HMAC key forever?


That is seldom a good idea, due to the "break once run everywhere" scenario.
I would instead use a Key derived from a Masterkey. The Key is on the database and the Masterkey on an seperate isolated device. From time to time you should change the Key by a new derivation form the Masterkey and some well known data.  


  Would an asymmetric or AEAD solution be preferable?


An asymmetric solution makes less sense here, because you have to store the private key on your database system anyhow.