@Ethan Fischer my old friend! Reach out and we can chat more. Sorry for the late reply and hope it's still helpful. First off, keep in mind that you can purchase your new home with 3-5% down, so using home equity for the down payment might not make a lot of sense but you'd want to run the numbers with your lender. You can use the STR income for qualifying as long as you claimed the income on your taxes.
Is there a reason you think you'd be selling the property soon? If it cash flows and is in a good market, why not just plan to keep it? If you have it long enough, capital gains won't kill you because you will have made a ton on it. It's really a matter of what your long term strategy is with the property. I can't say this from experience, but I have heard SO MANY times that people wish they had just kept that property instead of selling it. If it works, keep working it and keep it and worry about the capital gains down the road whenever it's time to sell.