I think you should most definitely prep for your SAT every time you take it. You pay around 50-60 dollars per exam, and it would feel like a waste of money to me if I didn’t prep at all and then did poorly. I started taking SAT/ACT exams Junior year but in all honesty I didn’t study as much as I should and I regret that a lot. My scores I received junior year were good enough for my instate schools but for my dream school I had to take tests Senior year. Thankfully everything worked out but I was for sure much more stressed. If you think you’re busy junior year just wait until senior year. I was in the same place as you were junior year. I said " I have time senior year to do it, I’m busy now", but trust me you are much busier your senior year. On top of managing your classes you also have to fill out all your applications, write a bunch of essays/ supplements, and fill out scholarship applications and financial aid forms.
If I could offer any advice it would be to try to finish all of your testing Junior year if possible. Sign up for tests starting 1st semester of Junior year (preferably towards the end of the 1st semester, so you have time to study) and then continue to take them second semester as well. I’m not saying to take a whole bunch if you don’t need too, but take as much as you need to get the score you want. If you find that you need to take an extra test or two Senior year there’s nothing wrong with that. I’m just saying I wish I would have finished testing junior year to have one less thing off my plate.
Also I would recommend taking one SAT and one ACT. Whichever one you score better on focus on that one and raise your score.
As for studying if you study for 15 minutes or so on weekdays and then do full practice tests on the weekends for a few weeks (preferably more if possible, but I get how busy you can get once school starts) you should be fine taking the test. Also if the school offers any free SAT make sure you take it, as I stated above taking these tests can become very expensive if you don’t prepare yourself and find yourself not improving.
I understand how your school wants you to take your time and prepare but, spring of Junior year seems way to late to just start the process, especially if you’re considering schools that are hard to get into and very prestigious.