I could still access equity through the HELOC for future expansion and the rent would cover HELOC interest payments while I stabilize future assets. 

Price, Condition, Access. 

Bridge cash out is a great option, especially if you cannot access the property bought at auction prior to closing. 

Yours has far easier access. 

I'm a Real Estate Licensee who works in a very remote, fly-in access-only community.

I can see a scenario where I take on the risk of a high interest rate and less flexible terms for 5 years, so I can access my cash sooner if that option is there.

- all you would do is "manage the manager", which you could do from anywhere you have internet access.

What is the fastes legal action to take in NJ to get access to the rental house? 

You could also request it from the agent who quoted that carrier on your behalf since they have access to those carriers / underwriters directly. 

Example a few weeks ago: got a "customized' postcard saying something like "Dear Andrew, we noticed you recently bought a property on 3455 Main Street - we have access to many properties just like this one, all over town.