If you have a serial column or an integer one that's automatically populated with a nextval (so that you are never supposed to insert new rows with an explicit value for that column), you could additionally check whether the value of that column is greater than a specific value:
(
(("qb_id" IS NOT NULL) :: INTEGER +
("xero_id" IS NOT NULL) :: INTEGER +
("freshbooks_id" IS NOT NULL) :: INTEGER +
("unleashed_id" IS NOT NULL) :: INTEGER +
("csv_data" IS NOT NULL) :: INTEGER +
("myob_id" IS NOT NULL) :: INTEGER) > 0
OR
YourSerialColumn <= value
)where the value should be determined as currval of the column/corresponding sequence at the time of altering/recreating the constraint.
This way the IS NOT NULL checks will apply only to the rows whose YourSerialColumn value is greater than value.
Note
This can be viewed as a variation on David Spillet's suggestion. The principal difference lies in the fact that this solution does not require structural changes (partitioning). Both options, however, rely on existence of a suitable column in your table. If there is no such a column and you can add it specifically to solve this problem, going with the partitioning idea might be the better option of these two.