The process of part exchange is not available on the scheme. However, if the dealership deem the car suitable, they may be willing to buy it with a view to them selling it on. If it’s old or not in great condition, they will usually try to find a dealer willing to take it on, and will only pay the price they’ll get at best. Some dealerships simply won’t bother, at which point you can either go with We Buy Any Car or any other similar offering, then use that money for your AP.
I think the dealer saying they don’t take Part Ex is a simple case of they can’t be bothered.
I'm Autistic, if I say something you find offensive, please let me know, I can guarantee it was unintentional.
I'll try to give my honest opinion but am always open to learning.
Mark