Unique value tells tjat all values in recors have idfferent means no duplucate value.
So just make a script and run the step like setfield/replace field
make sure the value already does not exist in that field.
Also 0/NULL is also the part of unique key.
if 5 is there u can't add another 5 in that field
so keep going in this way.
"Or is there an easier way to clear out the registration field for the new year?"
I suggest you don't clear last year - you would lose your records of the prior year's registration. Instead, why not just continue adding Registration records but include a year or date field so you know one year from the other. If you blank/change/remove the table's unique serial ID then you defeat the purpose of it; and it cannot be used to relate to Contacts or any other table.
The registration field needs to be cleared because it is assigned by the order in which patrons arrive. I maintain last years registration number to know who participated last year by copying the current number into a another field (last years registration number). No other field is cleared so I don't lose any information, just the order in which they came in the door.