Go to the children's current school and obtain copies of their transcripts. All schools ever attended will be there, with dates. This will clear up that extra year and tuition issue as a yes or a no.
Get copies of everything the father has paid. Hard proof he paid it.
Present these valid proofs to the Mother. She can argue with facts all she likes. Or even Dad can, if he's actually wrong.
Point being, bickering has no place in CS. Facts matter more, and if you have proof of 3 years and not 4, it's a non-issue.
Dad will be paying what he was ordered to until another order replaces it. Period, end of conversation.
Even if you don't get to the custody change for another year. He may as well accept that and move forward, so that everyone can.