I definitely wouldn't recommend asking for a signed and dated letter. It probably doesn't change your rights. Your vesting is determined under the plan and there may not be much discretion. I wouldn't expect to receive the letter, so you could be asking for something they won't give. Such a letter might be helpful in explaining yourself later, but you could also burn bridges now. It is aggressive and implies litigious intent, and it might broadcast that you plan on leaving.

I would instead ask HR what has to happen to get the system fixed so your vesting percentage is right. You can call back multiple times (across weeks or months) until they fix it. The HR person may not like hearing from you, but you will probably get a result.

No matter what the system says, if the plan document says to do XYZ for vesting service and you do it, then you have vesting service. The administrator might not process your payment correctly at first, but you aren't abandoning any rights by not getting a letter.

If you leave the job and they process the payment wrong, explain to them that you should be 80% vested and give the reasons why, then let HR sort it out. If they refuse you, you can follow the complaint and appeal procedures in the plan document or SPD.