The official system requirements state that SQL 2016 is compatible with AX 2012 R3 CU12 or higher.
You should not experience any problems migrating your DB from a lower to a higher SQL version. But do know that once the database has run under the newer version it’s impossible to downgrade! You cannot restore a backup from a higher version on a lower one.
Keep that in mind, if you have your TAP environment with one SQL server.
I don’t see anything wrong with your planning. You can also swap step 2 and 3, so that first import database on TEST and then upgrade SQL. I haven’t seen any issue for minor upgrade of SQL (sp2 to SP3). But I would add that as you are currently on R2-CU7, you need to take care of few things such as DIXF, you need to remove that from installation as DIXF is available in standard R3 installations.
Also I have seen DIXF service to crash with few versions of SQL. I can remember SQL server 2012 SP2. You need to watch how your SQL version will be at supporting DIXF service. Test it well.