You must deny the claim on the basis that BUPA has incorrectly calculated the payments and that they have paid you the correct amount. Therefore, the failure is on the part of BUPA, and you are not liable for the amount claimed.
If BUPA claims the monies as a debt, you must deny the debt and state the monies claimed are a disputed damages claim. BUPA will have to issue contested Part 7 Court Proceedings for damages for debt and/or breach of contract plus interest and legal costs as opposed to following the undisputed debt recovery route of serving your company with a Statutory Demand for repayment in the debt in full within 21 days prior to filing a court petition a Compulsory Winding Up of your company.
Thereafter, you are in a bit of a tricky spot. If you refuse to pay up, the supplier may pursue you for damages through the Civil County Courts. However, the onus will be on the Claimant to enforce their legal rights against you. Whether or not they take any action is up to them. I cannot advise you directly upon whether the Claimant will in fact take any recovery action against you as I do not know what is in their minds.
In addition, the amount claimed of £3,839 is comfortably within the Small Claims Track threshold of £10,000 damages. Even if the Claimant issues Court Proceedings against you for this amount, plus interest, and wins at a fully contested Trial, they are unlikely to be awarded their legal adviser’s costs of the Claim. Legal advisers’ costs are generally not recoverable in the Small Claims Court: only Court fees for issuing the Claim and for the Final Hearing.
Nevertheless, the Claimant may be well-versed in such low-value recovery action against defaulting customers or may be able to instruct legal advisers for a low fixed fee to litigate the entire Court Proceedings on its behalf making it economic to claim against you.
I suggest negotiating a settlement of the claim on mutually agreeable terms with the Claimant.