My lender has said that Airbnb will count towards my income as long as I have a two year track record. I know you said that you don't have very much equity, but do you have at least two years of Airbnb experience on your tax returns? This might be able to help you with conventional financing. I also co-host several other Airbnbs to increase my income stream from this, which looks better in the bank's eyes.
There are some other options you could do:
1. get a cosigner - family member, friend, colleague, etc. They don't put any money down, and you pay them for the use of their credit score. You can refinance after a few years and take their name off of the mortgage.
2. Banks and lenders are unfortunately more friendly towards traditional long term rentals. I've had a bank tell me that my Airbnb income doesn't count, but if I had a signed lease agreement it would. It might be worth considering long term rentals until you get portfolio up and running or build up more equity.
3. set up a line of credit with a bank. With your good credit and debt to income ratio, you could potentially qualify for enough to get a smaller house, and since this is considered "cash" you could get a better deal and wouldn't have to wait to refinance.
I personally haven't had very good luck with hard money or private money loans for use with Airbnb. The seem to cater more to flippers, and the ones I have talked to want their money back too quickly (6 months-year) which doesn't give me enough time to qualify for a conventional refinance. Not saying it can't be done or not to try it, but that has just been my experience.