I would use a carrot & stick approach:
if a member of the pool reports a news item inaccurately - or with a partisan slant - they are OUT of the pool. Period.
Simply put, when individual pool members start acting like journalists - and not as members of the opposition party, they can be admitted to the pool.
And when they bitch & whine, they should be informed that being a member of the pool is a privilege - don’t abuse it don’t be a partisan hack.
Perhaps White House press credentials for ALL the current “reporters” should be pulled and then be earned to be allowed back in.