Are the club permitted to call in the 4th official, who usually lives fairly local, to do an inspection early on a Saturday morning if it looks doubtful or are there League rules that do not allow that.
No, the League have to do that - and usually only do if the match official cannot make it to an inspection for whatever reason (work commitments, location etc). We can ask for a pitch inspection, but they (and postponements) are down to the match official and the League and NOT clubs.