As drsox says, dispute the claim and use your own documentation as evidence in response. I can't recall if you did this, but if possible get confirmation of your service activation/migration date from the ISP that you moved to after leaving Supanet as additional information. If they still disagree then request a deadlock letter and give them 14 days to respond or drop the claim. If they continue to disagree then take it to the ADR.
It's worth pointing out to Supanet that charging full price for a service that has not actually been taken / used (phone and broadband), because you moved provider, may be considered an unfair term by Ofcom. Likewise Ofcom does not allow auto-renewing contracts, so if they stopped taking payments past the original end date and still renewed the contract then that is unlikely to hold much water in the eyes of Ofcom, ADR or a court of law. Particularly as they didn't pursue it until years later.
Check Ofcom's general condition rules.
www.ofcom.org.uk