We had similar case where the DMK being used within the DB to protect some columns in the database, and DMK protected by Server Master Key (SMK), this is good as always the DMK is opened as long as the SQL Service running with same SMK, there is no need of password in which case the application need to explicitly
CLOSE DMK using password.
I agree with:
You should make all the nodes in your Always On Group have the same service master key, then failover would be seamless. https://learn.microsoft.com/en-us/sql/t-sql/statements/alter-service-master-key-transact-sql
Having same SMK across the replicas in availability group would be seamless. It may considered at some-point as less secure, well we still maintain the boundary of AG. i.e. the database cannot be functional when it is restored into a instance other than AG replicas, and we need to extend that boundary (one instance to multiple) as we need to compliance to the High Availability.
Consider it like maintaining same SID of logins across Availability Replicas. To make SMK same in all replicas, you can do following (if there are no other things protected already by SMK on secondary):
---- On Primary replica
SERVICE MASTER KEY TO FILE = '\\Sharedfolder\SMK-202005121328.key'
ENCRYPTION BY PASSWORD = '*************';
---- On Secondary replica
SERVICE MASTER KEY FROM FILE = '\\Sharedfolder\SMK-202005121328.key'
DECRYPTION BY PASSWORD = '*************';