Maybe. My LSAT wasn't very good and I still transferred up multiple tiers. I started at a tier 4 as well. I don't think LSAT grades are very important for transferring. Easily the most important are grades and class rank.
The answer, as always, is "It depends."
In the majority of situations grades and class rank are the dispositive factors. If you have very good grades and a high class rank you can probably transfer from a T4 to a T1. The thing that transfer applicants have to keep in mind, however, is that terms like "T1" and T2" describe a broad range of schools with varying degrees of selectivity in admissions.
For example, if you're trying to transfer from a T4 to Stanford, I think they might very well take your LSAT into account along with class rank, GPA, and the reputation of your current law school. The competition for those few transfer spots is going to be fierce. (I know someone who transferred from Hastings to Berkeley, both T1. They were ranked near the very top of their class after 1L, and had an impressive resume to boot).
OTOH, transferring from the local T4 to the local T2 (or even a non-elite T1) is going to be based largely on GPA/class rank.
One thing the OP may want to consider is whether transferring makes much sense in the first place. I'm not sure that any benefits gained from transferring to a California T2-T3 are necessarily much better than those gained from graduating top 10% at a T4. In fact, I recently worked at an office where a top 10% T4 grad would have been viewed as highly preferrable to an average T2 grad.
I went to a T3-T4 in CA, and the top 10% of my class secured good jobs. Some went to Biglaw, others to government/judicial clerkships, others to very reputable boutique firms. If you transfer, I assume there is a good chance you rank will drop. Also, I believe transfers are often not eligible for law review. Things like that may or may not
be compensated for by the increased tier rank.
Just something to think about, good luck!