Gawd, job hunting is so demoralising. I know the feeling.
I am bank staff dinner lady, it took 8 months to get my first shift but now working about 25% of the time (I suspect there's a cost savings for the school in not having to make me a regular employee ).
If you could do that, it would get you a recent job reference and possibly be a way to get your foot in the door of same school. What about another way of working as bank staff or for a temporary agency? Don't just think primary, look at secondary school bank staff, too (including dinner ladies).
You will need to consider childcare for your DC unless you can work at their same school (I know this is a pain). Also LSA jobs are sometimes more limited hours, something to check on.