This is sounds like one of those silly puzzles where the answer isn't what you would expect - whats a half divided by a half being my favourite
Lets go back to first principles here
Are you billing the "the third party" for your services ? i am presuming you are?
If that is the case it's fairly simple.
If you are no vat registered you bill the third party what you expect to be paid and they pay you that amount - if they are not paying you that amount you need to discuss that with you as there is come confusion as to how you are calculating your bill.
If you are vat registered unless your customer or you know of a reason why not you should add vat to your bill - the amount you charge before vat will be as agreed between you and the third party on the proviso vat will be added to that calculated vat and be paid across. The vat shouldn't be the issue here - its agreeing the net value to be billed that could be the issue.
We don't charge you VAT.
It seems like they are stating the obvious here - its reasonably obvious (to me) they don't mean what they say here despite being correct in what they are saying.
I.e I am presuming you are billing them and they aren't billing you anything - that being the case they obviously won't charge you vat as they aren't billing anything - so their comment is somewhat nonsensical unless i am misunderstanding the how the setup should work - it is possible they are just talking gibberish as they may be as confused as as i am