I hope were not splitting hairs here but there are three issues to be considered.
1) If the van is to have an extra seat installed in the back, then that work has to be certified.
2) An extra seat will draw a VRT charge of 13.5% of the open market selling price of the vehicle
3) I'm not sure why you are mentioning that the vehicle tax class is changing to private. The OP hasn't mentioned that and until he does im assuming that this is a commercial vehicle being used for commercial purposes "with a seat in the back" and one which continues to be taxed commercially. I'm not sure if you are implying that an extra seat in the back triggers a compulsion to tax the vehicle privately - That is certainly not the case. Taxing the vehicle commercially or taxing it privately is dependant on what the vehicle is being used for and not a function of the number of seats !!