How to do an UPDATE for all the fields that do NOT break a constraint?

I wonder if this is an SQL limitation or something I'm missing in the
PG manual, but I need to run an update on my database (to replace the
value of a column to match a new design structure).

Due to the new business logic, the  replaced value of a field may end
up being already present in the database in another record. This leads
to unique key violations when I run the update.

My question: I don't mind if the update transaction skips the records
where the key would be violated (this preservation is in fact what we
want) but these are only about 2% of the overall updatable records.

Is there anyway to make the transaction go through with the remaining
98% of the update SQL which will in fact NOT violate the unique
constraint?