Based on my understanding of the structure of the PSC, the answer is not very much at all.
There are three key documents. The first is the Articles of Association of the PSC, the second is the PSC constitution and the third is the constitution of local branches (examples can be found here, here and here).
Local branches are almost entirely independent of the PSC according to 11(b) of the Articles. Their members are to be encouraged to become members of the PSC (see section 3 of the branch constitutions) but need not be. The PSC constitution does try and ensure that at least the officers of local branches are member of the PSC but it doesn't appear to be entirely successful. The relevant clause is 5.3(b) which says that branch constitutions must include:
A requirement for elected officers to be members of PSC.However, the constitution of the York branch, for one, doesn't appear to contain any such requirement. How many other branches are in breach of this?
So what control does the PSC actually have over a rogue branch? The only thing they can do is disaffiliate, according to 4.5 of the constitution and 3(h) of the Articles. And the only thing that comes from disaffiliation is that branches cannot use the PSC name or logo.
The PSC is powerless, in effect, to control its branches. Disaffiliation is such a major step that it would surely only be used in extreme circumstances. And even if it is used, the local branch retains all the money it previously raised while using the PSC name and its members are still members.
It would appear that even if the PSC did decide to get serious about tackling antisemitism and Holocaust denial in its branches it really can't do anything without effectively closing itself down and restarting. It may be time for those who are genuinely opposed to racism to start considering that option.