We had the same problem. IIRC, we resolved it with our online account sign in to change it, but I may have actually done the supervisor route. The single name thing is their policy, unfortunately. We had the same problem with them, and I am 100% sure even a supervisor won't put it in both of your names.
Remember, you can use things that some people don't realise:
Council communications/bills
Letters from MPs
NHS/private doctors/dentists bills, notices, etc
Bank statements
water bills
TV tax licence
Gas
Electric
Loan statements
credit card bills
Mobile bills
Write to your MP a couple of times and (if you're lucky) you will get a letter addressed to your house. I think I used one of mine from our old address for ILR. I wrote to my MP here once and didn't receive anything back. You could write to the council and ask them to send you information on stuff. Or write a question and get a letter back. Not that I recommend using those stupid pay on the never never catalogues, but I am pretty sure if you charged something cheap, you could use one of those statements for evidence.
Also, people should keep collecting evidence until they get citizenship. If you have to replace your ILR if you lose your passport, you will need to show letters to replace it.