LPS, Local Day School or try again for 10th grade?

Totally true, for both LDS and LPS.
But it’s hard to give something a true try when you are starting to gather recommendations and write essays a few weeks into the school year.
so that’s why I say: if you are planning to apply to BS for next year, then I would probably not do a new LDS, assuming there is some familiarity with the LPS system already.