This sounds like the start of a negotiation to me.
It is good that you have established the principle, and that they now accept it - the argument now is over the amount.
I can't remember the original question - indeed I may not have even contributed to it.
Are you still employed there?
Have you got anything in writing that offers the £500 (and hence accepts the principle)?
Have they pointed out the document where the 'three month policy' exists, and supplied you a copy?
If you are still employed there, it could be worth talking to ACAS. If this is claim for unlawful deduction of wages, you may be able to take your case to an Employment Tribunal.
It is unlikely that the company will wish to do this, but you need to marshal your knowledge and resources of where this could lead them and you to, as a means of getting a better offer out of them.
You do need to be clear to them about how much you are claiming (£2k or £3k) because it will make a difference to the way they think about their 'start point' of negotiation.