How do I use a different domain and username in a SQL Server BACKUP LOG command?
Powershell won't help, it is just a wrapper for the BACKUP command. Your problem is that it is indeed the database engine that produces the backup file, and there is no option for impersonation here. So, you really need to sort out that service account, I'm afraid.

(There used to be something called a nullsession-share. I doubt it still exists, and basically you open that share to the whole world. Not recommended, I'm just mentioning here to recommend against it.)

Come to think of it, how about the computer account? I'm not sure if that will help, but if you grant privileges to that share for the computer account where your SQL Server is?