Since they get a lodging premium from their employer, as Adam said they will pay more than your standard tenant. 

No DTI, tax return, or employment checks.

Back story is, I’ve done the unthinkable and relocated with no housing or employment lined up, however, I have a very strong work history.

In this case, if our girlfriend is self-employed and not doing business through a Corporation and you paid her more than $600 a year, then you have to file a 1099 form for her.I recommend to consult your accountant to figure out if your rental activity is counted as a business or as an investment.

You are not required to give more details and, just like an employer (and for all the same reasons), you should not.  

I have 3 rentals now and the banks aren't very keen on giving me another mortgage with my self-employed income. 

First, look at lists of cities ranked by employment, employment growth, net in-migration, cost of living and other economic factors and find the top 3 on those lists and then start looking at properties and neighborhoods in those cities until you find one that grabs you.

There is some variability in the different neighborhoods.The best towns to do a house hack in my opinion are West Hartford, Newington, Glastonbury Rocky Hill, And generally any good Suburban town outside of the working markets.Primary employers are clustered along routes 91 and 84 and the Merritt Parkway.

So my first question is this: I'm using QB Self-Employed for our personal finances, would it be a good idea to simply invoice each business for my time that month?

Don't give up, you're in a great area that has seen high growth, almost zero vacancies, and is supported with diverse employment sectors and NJ and NYC transplants