If so, great, otherwise you can just as easily 1031 in to a nice property near you (probably bringing in less rent but still no debt.) or you could use it for 1/2 down on two properties that would probably rent for more combined and have a low/manageable payment. 

I have been approved for a 390k conventional mortgage as well as a 375k FHA.

You should hire a third party person that is approved by the seller like your title company or an attorney or any individual you both can trust. 

I got approved for an FHA loan, which to my understanding doesn’t let the owners rent out the property.

I am a 1099 employee, which is making the pre approval process quite difficult.I have great credit, no debt, and sufficient funds, but since I don’t have 2 years of 1099 income that doesn’t seem to matter.

But I am still operating with the mindset I had when I got the first house - basically get pre-approved first and then look for the best house you can afford.

I was pre approved for 175k.

Essentially you have to prove to the loan underwriters that you will be able to occupy the home within 60 or so days or they won't approve the loan.

We're waiting on the lender's final approval to close on #4 (80 units), and now have another 150-unit under contract. 

The house was just listed on the MLS stating it is an approved short sale at 260k.