SSAT At Home maybe?
Also, there’s a program, I forget what it’s called, in which you can take the SSAT essentially privately. You can do it once IIRC. You pay extra, but there are more test centers offering this than the bulk test centers. Maybe one near you. And you can arrange date/time as you like, assuming they’re open at that time.
BTW I’d add that your current school is for sure going to have to know and participate in your transfer process. You’re going to need letters of recommendation from current teachers. Given that you’re going to have to tell them, maybe they’d give you some sort of grace to skip classes one Saturday to take the test.