Shaun,
I don't know what type of HA you have implemented but it sounds like perhaps it's an AG with two servers and a witness? My experience has been that it is better to run the DB installer on both servers. There are lots of .net objects, sometimes additional required SQL packages, etc..., and if you are in an AG vs an FCI it's a pain to replicate all of the SQL Server Agent jobs (many of them change). We usually do the folllowing:
1. Suspend data movement on the secondaries
2. Remove AW (WS1 UEM) DB from AG on secondaries
3. Turn off SQL Server Agent on the primary and secondaries
4. Take a full backup (we do log backups every 15 min so this isn't necessary but it doesn't hurt). The backup is done with the ' copy only' option to prevent logfile/backups from getting out of order. If your DB is large a multipart backup will help tremendously. I have found a 4 part backup to be ideal for us.
5. Take a log backup.
6. Run the installer.
Once the DB installation is complete work on other servers proceeds and I then run the program on the secondary
On the secondary
1. Drop the DB
2. Create empty blank DB as outlined in onprem installer guide.
3. Run the installer
4. Drop the AW DB again when the installer is complete
5. Restore the full backup from the other server to the local one with the ' no recovery option'
6. Restore the log file backup from the other server with ' no recover option' . I have sometimes had to take another log file backup from the primary and restore on the secondaries
7. Rejoin any/all secondaries to the AG. If you are using TDE this would have to be done with T-SQL there is a bug in SSMS that prevents you from doing this using the GUI once TDE is turned on.
8. Verify the AG/HA solution is working as expected.
9. Restart the SQL Agent services on all servers.
Hope this helps