It seems you've done everything you can that could be considered reasonable and appropriate to ensure the fine is paid. Copy the phone numbers, dates, times, and call durations out of your phone's call log, make a 1 line summary of each call, store in a safe place and then forget about it.

If it pops up in 6 months time, just send them a letter with a copy of the above and an indication you're still happy to pay the fine given an appropriate method of doing so.

At that point if they try to stick you for late fees, well, I love dumb standoffs like that. I got into a spat with Three in November 2012 that took until 2017 to resolve (about 2 letters sent/received each way per year). Eventually had the £70 "debt" written off, simply because I (rightly) contested a £20 component to it.

They never provided a reasonable method to pay the £50 part despite my repeated requests. It was a boring process but I thought a very useful lesson should a situation like this involving much more money ever occur