Short answer: yes, you can certainly do this if you want to and the result is valid. The main downside, in my opinion, is that the backup is then dependent on more technology, so there are more things that could go wrong. More on that later.
If you're going to to this though, you really don't want to end up with Online Backup as your backup solution. The problem with online backup is not consumption of resources, but time to restore, I thought you were going to say you wanted the DR system so that you could shut it down for a couple hours while you take a cold external backup. That would be a pretty good reason to do this.
Since mirrored databases record their journal location inside the database, they intrinsically know from what journal file they need to "catch up" (the mirror checkpoint info). Like all the usual backup solution, the result is not transactionally consistent in and of itself, but requires journal restore following backup restore to get to a transactionally consistent state. Mirroring makes this easier via the aforementioned checkpoint and the automatic rollback as part of becoming primary. Of course it's the mirror journal files, not the DR's own journal files that will be used for this, but they live in the same directory, so if you just back that up in the same backup, you'll have the right stuff if it ever came to restoring this.
Now more about those downsides. Backing up a replica means that you are subject to any problems with the replication. For example, if a database on the DR had a problem and we had to stop dejournaling to it, that could mean your backup isn't good. You'd worry a bit that you didn't notice because nobody is running on the DR system. Or if you add a database to the primary but forget to add the same to the DR, your backup wouldn't have it. These aren't meant to say this is a bad idea, but it is a consideration. You want to think a bit about what you're trying to protect against. You're talking about having a DR, so if you're restoring backup it means that something went wrong with both the primary and the DR. So is the backup of the DR good in that situation? If both are in the same physical location and your backing up in case that location is destroyed, then you're protected. Or if you're backing up to handle the case of errant/malicious deletion of data, then you're protected.
I don't know what your situation is with the main server, but I'd be curious how the system architect expects backups to take place and how long a backup of the disks are expected to take. With a large global buffers, ExternalFreeze() can be workable in some application environments even if the freeze will last many minutes. If your operating environment is such that good backups are an absolute must, you might be better off investing in getting external backup working over there.