Ohhh, yeah that's an odd combo.
I would suggest seeing if you can rent an infrared analyzer this winter. I did it 2 years ago and again last year and will do it again this year, worth every penny. I crank the heat up to maximum to get the house as hot as possible, then I turn it off and start analyzing heat patterns and look for leaks. Every winter I am sealing 10 leaks I didn't know about, and the difference is night and day -- I think I lowered my heating bill by 30% the first year and maybe another 15% last winter. This winter I already know two spots on a wall that are cooler so I have to figure out a way to insulate them, and the bedroom ceiling in my master suite also appears to have cool spots so into the attic I will go to figure that one out.
I know the previous owners of the house (they lived in it for 2 generations and about 40 years!) and they're stunned by how low my heating bills are and I showed them infrared photos I took of all the leaks they had for probably 40 years. They were paying MORE than double what I pay, and I keep my home 1 degree warmer than they do.
Either way, do some research for sure. While a HELOC can be had RIGHT AWAY, you may see significant or even huge savings just by working on heat loss abatement first. I supplement my boiler with electric heat in two rooms (bathroom space heater and a electric foot heater in my bed) and it only adds maybe $5 a month to my electric bill.
