Add INCLUDE term to primary key index in Postgres 11
Without messing with the catalogs (which is not commendable), the only option I can think of would require you to do without foreign keys for a while:

You can define a second UNIQUE index on the table that contains the appropriate INCLUDE clause.

If you use the CONCURRENTLY clause of CREATE INDEX, that shouldn't be disruptive.

Then you can delete the original primary key constraint and all dependent foreign keys using DROP INDEX ... CASCADE.

Then use ALTER TABLE ... ADD CONSTRAINT ... USING INDEX to turn the unique index into a primary key constraint.

Now you can re-create the foreign key constraints.