Before you start tearing the bulkhead apart check that your mast is centered and confirm your tensions with a Loos gauge.
To make sure the mast is centered I run a 50' tape up with the main halyard. I then measure to each outer chain plate. If it's not the same then I adjust accordingly until it is the same.
Next I try to get the mast straight. That's basically a case of adjusting the lowers until the groove appears to be straight. Running the halyard to the base of the mast and pulling it tight will give you something (fairly) straight to use as a guide. Once the mast is straight, use the loos gauge to adjust the tensions to match the manual or tuning guides. If you've done it right the mast is still centered and straight, and you're good for . . . . well . . . a few days or weeks.
Your rigging will stretch over time, plus the boat will change shape a bit as it adjusts to being in the water instead of on a trailer. It's not unusual for me to adjust the tension 3 or 4 times during the summer. I'll generally have to adjust a couple times the first month, then maybe another couple of times during the rest of the season. It's normal for the downwind lowers move around more as the wind speed goes up; That's because the mast bends due to the increased forces. Half an inch of movement is probably ok, but if it's extreme then it's time to tighten them up (or reef!). You'll find that you need to tighten the lowers on particularly windy days. FWIW, I always seem to have a bit more flex in one side than the other, even with the tensions the same. Quirk of my mast, I suspect.
Anyway, my point is, make sure the rig is squared away before you start tearing into the bulkhead. If it is, and you still end up with significant slack on one side, and not the other, then look at the compression posts, chain plates, etc. Don't skip checking below the chainplates too, as I seem to recall a case where there was rot at the base of the compression posts and not at the cabin top.