A HELOC is not a guarantee they won't shut you down. I admit I've never read the fine print, but if banks have the slightest whiff of impropriety they can shut you down.
In my case, I had a HELOC at a bank that I was using for MO deposits. The HELOC typically has a 10-year term, after which you can't borrow anymore money but have to start paying it off, and my HELOC timed out on the 10 years shortly before they closed my checking account. They went so far as to block my internet access to even see my HELOC account! I had a high 5-figure balance (acquired in a way having nothing to do with MS) and of course I have to pay it off.
If you have good credit you can get a HELOC at around prime these days, but prime is climbing. Although you could probably make a payment BEFORE taking the advance, which would temporarily leave you with a credit balance, but you wouldn't pay interest. HELOC's usually have significantly lower closing costs than first mortgages.
I think you'd be better off opening bank accounts. When they shut you down, open another. There are lots of small local banks and credit unions.