be good-
Is the USPS paying for this out of pocket?
Try to get the letter from who ever the adjuster is that is settling the claim, if the adjuster isn’t authorized to write the letter ask who is.
I’m no expert at this but I think the your insurer would have more respect for documentation from another insurance company as opposed to the USPS.
You don’t have to give the letter to anyone, just keep it safe in case your insurer tries to raise your rates- you can prove that it wasn’t your fault.
On the slight chance that the other driver files a personal injury claim against you- showing a letter from their insurance company stating that you shared 0% of the responsibility should do wonders.
Just keep in mind that while the claim is still open you have the upper hand, make sure to get the letter before the claim is closed.