I would love to connect to talk through some options for this client and others. 

I also have a partner with a full time job and great credit score as well, so maybe my best option would be to include him and have him apply for the mortgage.

You will then have a few options with that rent money....you can put it into principal or you can start saving it to purchase yet another property.

@James Bakun, A simultaneous exchange is the only other option other than what @Melanie P. laid out. 

In that case, have you considered the option of using the HELOC to purchase the home & make renovations - while then renting it out for the time being, instead of worrying about renting out your current home (with flood concerns)? 

The cheapest option isn’t always the best.

Short-term rentals come with fixed expenses like licensing, listing fees, and optional property management and furnishings.

Other options would be an LLC owns both homes and you each own half the LLC.

Especially going the low down option you will get the best bang for your buck. 

It's wise to thoroughly discuss your options with the property management company, consider your needs, and ensure clarity on contracts, insurance coverage, and benefits arrangements before making a decision.