Hi Jason,
You explained your situation really well, so I am going to try and address each part.
1. $40-$50K is enough for a down payment for a house in Denver. Potentially even, if you and your wife are open to it, to get you a place with rental capabilities on site to help you off-set a good chunk of your mortgage.
2. You should definitely still be able to get an investment HELOC for minimal closing costs. I know of local places in Colorado, but unsure if they would work for a home in Georgia. I can send you some contacts to see if they know more. Terms will not be as good as a HELOC on your primary but should still be worth it. Unfortunately as others have said 1031 has to be for a "like-kind" trade investment to investment.
3. Is your house in the town of Fairplay or outside? That will determine the short term rental rules, if you choose to go that route (which will help your cash flow, but is more work). Inside town limits has its own rules vs. outside follows Park County rules. I have property in Alma and prices are still going up, where are you getting your home valuation from? I would be happy to run a CMA (Comparative Market Analysis) just to be sure on valuation. I agree that the rental situation could be a seasonality issue and you might want to consider a 3 month lease for summer and 6-9 month for someone that wants to be out there for the winter months. Although, most lenders will only be able to use a 12 month lease to help you qualify for your next property so that is also something to take into consideration on your search.
Hopefully that is helpful! Always happy to chat further
Best,
Stacy