Does the Mass have to be said on the allotted day?
No it doesn’t. Clearly it will not always be possible to arrange to have a priest say the Mass on the allotted day. He might already have to say another Mass on that day. You don’t then have to find another priest. Just have it said on the first day that is available to the priest either before or after the allotted day. Remember that whatever day the Mass is said it is still said on account of your allotted day so the connection will always remain.