If you got your FHA mortgage in 2018, paying it down to 80 percent won't make the mortgage insurance go away. You'd have to refinance to a different loan program.
If the house that your mom lives in is in her name, you have a couple of options. She could tap the equity with a reverse mortgage (assuming that she's at least 62 years old) and you could use that money to do a "cash-in" refinance to an 80 percent conventional (non-government) mortgage for your primary home. Of course, reverse mortgages also have mortgage insurance, but you won't have to pay it out-of-pocket. Your Mom might even be able to get additional cash if she needs it, while being able to live in the home as long as she wants with no payments except taxes and insurance. To get the most money possible, you'd probably want to get off title to that house. But you don't have to. It's just that the loan amount is based on the age of the youngest person on title. https://themortgagereports.com/23246/hecm-reverse-mortgage-who-should-consider-it
Other option is a cash-out refinance of your Mom's house, again using the cash to do a "cash-in" refinance of your house. But you'd both have to qualify for the refinance because there would be payments involved.
As to which banks do HELOCs on second homes, I'd look at local small institutions, credit unions if you have access to one, or specialty lenders. Other option is to apply jointly with your mother for a HELOC against her primary residence, with you as a co-borrower or co-signor. https://themortgagereports.com/31835/short-on-funds-get-a-co-borrower-to-help-you-purchase-that-home
Thank you for writing, and good luck with your search.