To do this I make agreements with tenants to extend their lease a couple of months as is and they allow me to do some prep/turnover work and find a replacement tenant before they vacate. 

Every landlord should have a method, that can pass legal scrutiny, to get rid of undesired tenants .Just do not do something stupid like indicate that you are not extending their lease because they have an emotional support animal.BTW we have a unit with a legit support animal. 

Does she/he have to be added to the mortgage in all circumstances, regardless of expected duration of stay?

Tenants should be held responsible for duration of lease until you find new tenant.

Regardless of the strength and duration of your relationship, MONEY complicates everything.

@Bruce Lynn thanks for extending your hand.

If you you're a good risk and have shown success with REI they may be willing to extend a line of credit or have other options to expand your portfolio.

A lot of value, I extend my gratitude!

It won't affect your DTI and your able to obtain the funding that you need in order to extend your portfolio.   

I have a background in General Contracting and I've extended my contractors to my investors who have been doing their jobs at contractor pricing.