I would definitely send a cheque, so they don't have your card details, and do check your records if you have or can get them.
I looked into this a few years ago, and unless the law has changed, they can chase you for up to 6 years, so they are within the time frame, even if it is lax.
I was contacted by an institution that claimed I owed them money 5 years after doing a course there. I knew they had been paid (it was a substantial sum to me, and I would have known if it hadn't been collected) but the bank archived records after 2 years and wanted a fee to search the archives.
I was moaning to a friend, and included the moan that the institution was being absorbed into another. He was an accountant and seized on that, saying 'oh, they're going through the records and tying up loose ends - just try writing to say that your records show it has been paid'.
I did and received a curt letter saying the matter was closed. My friend explained that they just needed my letter on record!