I agree with the other poster that you should check with your instructor, but I just finished PSAI cavern and here is what I did. I upgraded a lot of gear.
BC
I had always used a Transpak, and if you are doing cavern with a single tank, it is all you need. I also did Cavern in doubles, so I upgraded to a back plate and wing, also Dive Rite. I got a steel backplate and the Classic Wing, with redundant bladder. You need flotation redundancy in either your wing, or by using a dry suit.
Regulators
I already had done the IANTD Adv Nitrox class so I had three regulators, 2 - DR RG2500s and 1 - Mares Rebel, O2 clean. I configured the long hose (7 feet) on my primary regulator, and if using a single tank, you will use an H valve on the back gas cylinder. Your SPG will go on the backup regulator which will go on the backup post. My instructor had me put my primary on the right side, and the backup on the left. Again, this is what I did. Consult with your own instructor for his preference.
For a primary light, I used a DR 10W HID from rental. I had never used a canister light before, so I wanted to try it out first before I jumped into buying one. Now, I am looking at the Salvo 21W 10.4 A model, so I am glad I didn't just buy the 10W right away.
I had 2 backup lights, but for cavern you really only need one, as the sun is truly your primary light source. Check with your instructor.
I bought a Z-knife and placed it at the top part of my harness on the right side. I also had a primary reel for running a line into the cavern from the entrance, and a spool for doing lost buddy and lost line searches. Again, your instructor will explain all of this to you.
I used a 7 mil farmer john on my 3/2 neoprene wetsuit, but I believe I will make the jump to dry suit, as longer dives in 70 degree water will push that gear to the limit for me. If you are already dry suit certified, great. If not, consider doing that at some point. But you should not need dry suit for cavern. You can use a 5-7 mil out of rental, or maybe you already have your own. Use your own judgment for what exposure protection you think you will need.
If you are using split fins, there are some kicks that you are going to learn how to do for cave and cavern that are not possible with those. I am back to using my Mares quattro pros. Ask at your LDS about a good fin for tech diving if you have not already done so. It may be that the fins you are using now will work fine.
Your instructor may require you to carry a redundant mask. You will also need a pocket for your IANTD tables. I like them because they roll up, not like PADI slates. There is really not much need for a lift bag or a sausage in a cave. You can carry those if you are going to be in open water before you enter a cave though. Check with your instructor.
You need to carry some type of timing device or a computer. You are diving on tables in the course, so there again, you should have redundancy in your timing device, if nothing else. You also need to carry a slate. I use the one that mounts on your arm, and I replaced the cheap wooden pencil with the all graphite one.
I have not purchased tanks yet, but I am looking at double LP108s. My instructor uses HP130s and I find that that little bit of extra weight is too much for me right now. If you are going to use Aluminum tanks, be sure you are properly weighted. I recommend steel tanks for entry into the overhead environment. Again, check with your instructor.
I had the IANTD Cave manual as the PSAI manual had not been published yet. It was just recently rolled out at DEMA. Either gives you a lot of good information about what your limits are as a cavern diver. Read those. Also, if you can afford some extra, go on the NSS/CDS website and order the NSS/CDS Workbook. Read that. There is lots of good reading material out there for cavern diving.
I am trying to get a week planned to complete the rest of my training through full cave in the next few weeks. I will let you know how that goes.
Best of luck to you.