Are you asking if you are liable?
It turns really upon who was believed.
That cannot really be predicted.
They are alleging that you damaged the lock. You say it was already damaged. I can't really say which a judge would prefer. It depends on how you give evidence.
It is certainly right to say that they would not get a complete new door if all this needs is a repair though.
If the lock was already damaged then there is nothing probative really in the fact that they can produce evidence of damage.
Whether they will sue is another matter.
it is a very easy threat to make and much harder to enforce.
Can I clarify anything for you?
There is no offence of breaking and entering in the UK. What they mean is burglary and you haven't done that because you didn't intend to steal, rape or cause GBH
What it may amount to is a criminal damage though.
I wouldn't agree to anything at all.
Asking them for evidence of the 'damage'. It should be fresh if you caused it. It won't be. It will be rusted.
Why do you want to offer to pay for the damage?
But surely you didn't break it?
All you are doing by sending that is giving them evidence to use against you as they will say you are accepting responsibility.
I think I would probably write or text saying no more than that you don't accept causing any damage and that if they disagree they should sue.
All the best.