Can not having a hs diploma but a CHSPEE or GED instead backfire? Ideas for trasferring?

I’m currently a junior in hs. I used to have a pretty good GPA and was doing well in school, but I moved from private to public and all the way from LA to NY in the middle of the year while dealing with family stuff and my grades have gone down. I used to have high standards for a university, but I haven’t been able to focus on schoolwork at all and don’t think that’s a possibility anymore. I skipped two grades, so I am turning 16 the summer before senior year. Because of all that is going on, I’m thinking about not doing senior year and getting my CHSPEE/GED instead and then going to community college next year instead of waiting to graduate and then having to go to community college anyway. I would like to also do a gap year in Israel (possibly Bar Illan or seminary) and then transfer to a university back in the US. Would getting my CHSPEE and not a diploma be harmful in anyway and would I be able to transfer from Bar Ilan to a university (like UCSD, Cal State, etc) be possible?

Junior level transfers to UCs and CSUs do not need any high school stuff (high school courses, grades, and test scores), although some may be optionally provided to fulfill some base-level requirements. However, sophomore level transfers at the few CSUs that admit them will need high school stuff.

If you will no longer be a California resident, then UCs and CSUs will charge additional tuition without financial aid coverage.

Other schools that you may transfer to often do require high school stuff even for junior level transfers.

However, you wouldn’t be a CA resident AND you wouldn’t qualify for financial aid at UCs, and as transfer student you wouldn’t have access to merit scholarships.
My suggestion to you: On the contrary, stay in school, and take Dual Enrollment classes. It means you’ll get college credits for free and then will be able to shorten your time in college, but you’ll still be able to apply as a freshman.

" I used to have a pretty good GPA and was doing well in school, but I moved from private to public and all the way from LA to NY in the middle of the year while dealing with family stuff and my grades have gone down. I used to have high standards for a university, but I haven’t been able to focus on schoolwork at all and don’t think that’s a possibility anymore."

Run, do not walk, to the guidance office and get some help with the “family stuff” so that you can re-learn how to focus on your schoolwork. Until you get that back on track, you aren’t going to have the patience to prep for the GED, do well in your potential CC classes, or cope with seminary study.