I believe your idea is the true one, but to stand for Parliament as a Labour member you have to belong to one of the acknowledged factions to be sure of any support at all.

An independent member can count his votes by the capful." "That is the old system," she pointed out firmly.

"It is for you to introduce a new one.

If necessary, you must stoop to political cunning.
You should make use of those very factions until you are strong enough to stand by yourself.

Through their enmity amongst themselves, one of them would come to your side, anyway.