If you have 4k in savings and the banks are hassling for money that is owed to them, then I reckon you should pay up. If the 4k is in savings with the same bank they will be less lenient with you as they know you have the money there.
I dont think they will accept the 1k.
To put 4k on the end of the mortgage are you asking to increase the term by 1 year or to put the 4k on the whole of the mortgage whilst keeping the same term ? If keeping the same term, in the long run this is going to be very expensive for you and you would be better off just paying it off. In the future with the rates going up - can you really afford to let this small sum partially effect your ability to repay when the rates increase ?
Is the issue more so the fact that you are annoyed with how they are handling your situation ?