Recording baptism without a specific date

A member was baptized a few years again, but the baptism was not recorded. I know who baptized and confirmed the member and know of several witnesses of the ordinances, but no one knows the exact date of the baptism. MLS requires adding the exact date of the baptism in order to record the ordinance.

Is it necessary to know the exact date and is there a way to record the baptism without knowing the exact date?

Required fields in MLS are yellow. If the "day" field is yellow, it is required (and I'm sure it is). I know of no way around it. There is nothing clerks can do on their end to circumvent this. If it is indeed an issue, this must be discussed with CHQ. You can call them, message them through MLS or send them an email at msrmail@ldsmail.net

Review the Church Handbook of Instructions, Page 31.
This comes up quite often, so get familiar with it.
Basically, there are two options:
1) have the member or parents come up with two witnesses and ideally evidence that the baptism was performed, a certificate, program, photos, etc. and the exact date and who performed it and their priesthood office... OR
2) have the member be baptised again.