Hey Rob, welcome to BP! You're in the right place for sure. So, a few things to address here:
1. You'll want to make sure per your loan documents that you can move out and rent it without incurring any sort of penalty. Usually you can do this within 12-24 months with most loans. Likely you're fine there.
2. The lender is correct on PMI. If you can get an appraisal value supporting an 80% LTV (in most cases) then they will be able to drop the PMI. Definitely worth the cost if you think it will appraise.
3. This is where things get tricky. Without leveraging the existing equity in your home, depending on your cash reserves, it may be tough to get a new loan. You also may run into DTI (debt to income) issues because while you likely show enough income to buy the house, with the additional loan on the first house needing to be covered, it might be tough. Lenders won't be able to count "future" rental income to offset the debt, and usually want to see 12 months of this rental income on your taxes to include in your DTI calculation. I would speak with your lender and discuss what your options would be.
Good luck and reach out if you get stuck. I'm down here in Austin as well.