Does a letter fulfill the requirement of "..go and be reconciled with your brother"? Doesn't reconciliation always involve a face-to-face meeting?

The Scripture that comes to mind in your situation (in addition to those you have mentioned) is Romans 12:18:
In Romans 1:11-15 Paul tells the saints at Rome that he has desired to come to minister to them, but up to this point in time he has been prevented from doing so. And so what is he doing? Paul writes a letter to them. It was the best he could do at that time. I believe that this agrees with what Paul later says in verse 18 of chapter 12. We are to be at peace with one another, to the degree that it depends upon us, so far as we are able to do so. If the victims of your offense are unwilling to see you face-to-face then you should do what is possible for you, do what Paul did when he could not see the Romans face-to-face: write a letter. I believe that this does fulfill the passage in Matthew 5.