I could rent a nice class A apartment for about 2k-3k with a roommate.

I leveraged a lawn business & some retirement accounts to house hack a class a single family a mile from campus at 18 with roommates.

I started by buying my first home as an 80/20 (two loans, zero down) and had 2 roommates. 6 years later I moved out and rented it fully with expenses covered, had to rent myself for a couple of months before I could qualify for a loan on my next property, and then it really began!

Can tenant take on a (qualifying) roommate to help offset costs?

Personally i think its smarter to work thing out with the tenants I agree it's better to work with good tenants.Last time, as tenants consolidated/ moved with family or roommates or purchased with the 1st time buyer credit, we filled our B class units quickly with folks moving down from homes or A class apts. 

The other may not have been legit but it was already a unit with a pet rent and pet deposit so the roommate basically is avoiding pet rent but the initial tenants are paying a pet rent and paid a pet deposit.

I also had a roommate who rented a room from me, and covered most of my mortgage. 

They live alone because they could not keep a room mate.

It depends on if you are willing to also take on roommates in the unit you are living in. 

Have you considered moving in a friend and they're not so friendly German Shepherd as roommates to these pain in the butt tenants.