[akikuchi]

The reasonable letter is not intended to win sympathy, nor does it cut off other options. If they maintain a firm, unreasonable position, OP's options are still open and he can intensify the conversation. Starting with a thoughtful explanation of his side of the story is evidence of a good-faith effort to work out a solution, which improves his position if other parties get involved (e.g. lawyers, the internet).

Worst case: you're back where you started, but have demonstrated for the record that you were prepared to work together to find a sensible solution.

Best case: you've kept things civil, and come to an agreement without any inflaming the situation further.