Advice about Backup Solution
You could be utilizing the transaction log backups that are taken by the log shipping process.  In fact, you can't double-up on transaction log backups.

So, as long as you have the actual transaction log backups and the ability to copy them to other storage (in order for your custom backup preservation and retention for Disaster Recovery), you should be fine on the t-log front.

But, the real problem is that you can't take full (or differential) database backups.  You will need these for disaster recovery.  A file level backup is not good enough for a sound DR plan, and surely not sufficient for point-in-time recovery.  You must have, at one point in time, been able to take a full database backup in order to initialize your secondary database(s).  Why is that no longer an option?

BOL reference on Using Log Shipping as Part of a Recovery Plan