Okay, let's go over this point by point.
First, no Sony APS-C camera has a native flash sync speed faster than 1/160s. Full-frame Sony cameras can sync at 1/250s but only if you use a TTL-capable flash trigger, as this is right at the edge of their shutter performance envelope, and they need very precise timing for it to work - without such a trigger, you're still limited to 1/160s. One exception is the A1 - it has a brand new shutter mechanism that can do 1/400s, or 1/500s in APS-C crop mode. It also costs $6500 for just the body, and you need full-frame glass to go with it.
One way to get around this limitation is high-speed sync, which is fairly common in land strobes, but extremely rare underwater. You can easily Google an explanation of how flash HSS works, but for the purposes of underwater photography, your options are:
1) Seacam Seaflash 60D or 160D with Canon or Nikon cameras and wired triggering - while these strobes can be triggered over fiber optics, the high-speed sync option is only available over electrical wires, and only for Canon and Nikon cameras.
2) Retra Flash Prime X or Retra Flash Pro X (or, if by some miracle you find someone selling a used one, a non-X Prime/Pro model, but not the original Retra Flash with the straight flash tube) plus an LED trigger board capable of generating the HSS signal. To the best of my knowledge, only UW-Technics and TRT Electronics make compatible triggers, but they do support Canon, Nikon, Sony and Olympus/Panasonic in a variety of housings.
Keep in mind that HSS significantly reduces overall flash power.
Regarding battery life, I'm using a Sony A6300, same as you, and with new original Sony batteries, I was getting three dives out of a charge with 16-50mm or 10-18mm, and two dives with 90mm. Several years later, I'm getting two dives with 16-50mm or 10-18mm, and tend to run out of battery on the second dive with 90mm, so it's probably time to get a couple new batteries. I find this generally sufficient, but if it's not enough, you can probably get a small powerbank and stick it inside the housing - the Salted Line has a significant volume of empty space inside. An A6600 will give you much better battery life, but it will require a new housing and ports, and won't solve your other issues.
FWIW, I never use moisture absorbents inside the housing and I haven't had any problems with fogging during a dive thus far.
If you change to a camera without a built-in flash, you will need to use either an LED trigger or a wired connection. With the former, keep in mind that Sea & Sea strobes are known for having low sensitivity on their optical sensors - they're all fine when triggering off a camera flash that dumps lots of light down the fiber, and YS-D1s tend to be okay with LED triggers that aren't near as bright, but YS-D2s have been known to be troublesome in less than optimal conditions, and YS-D3s are even worse in that regard - Sea & Sea's own LED trigger uses unusually powerful LEDs and a comparatively large battery to work around that. Also, keep in mind that not all triggers are created equal - some are capable of TTL operation, while others are manual only, and HSS support is far from common. SeaFrogs have a very affordable (just $20) trigger, but it only fits their A6600 and A7S III housings, as it mounts on the camera's hot shoe, and these housings have LED portholes lined up with said shoe.
Alternatively, if you opt for wired sync, you need strobes that support it and a housing with the appropriate bulkhead. If you want features more advanced than simple triggering with manual power setting on the strobes, you also need a trigger board - TRT triggers are optical-only, but UW-Technics boards support both optical and wired triggering. One exception is Seacam Seaflash strobes that have TTL circuitry built into the strobe itself, but this is limited to Canon and Nikon, and in older models, it's Canon or Nikon, as they have separate SKUs for different camera systems. You also have to deal with fairly thick and bulky cables (in comparison to fiber optics), and multiple connectors that are secured with o-rings, all of which are potential leakage points.
The magnet trick doesn't require any modification of the housing - the pop-up flash has a small magnet in it, and the camera has a Hall effect sensor that is triggered by this magnet when the flash is down, letting the camera software known that it has been retracted. If you trigger that sensor independently, it will achieve the same effect, so you can get a pair of neodymium magnets, put one on the inside of the housing, the other on the outside, and use the outer magnet as a handle to slide the inner magnet close to the flash to disable it, and away from the flash to re-enable it. Note that if you use a trigger or wired sync in lieu of the pop-up flash, you can disable it from the camera menu (function - flash - off).
By 12mm Rokinon you mean the rectilinear one? The 4" port is too small for it - I mean, physically it might fit, but it really needs a six or eight inch dome to work anywhere close to properly. The four inch port is meant to work with fisheye lenses like Sony 16mm + VCL-ECF combination, or Rokinon 8mm f/2.8.
Edit: If you really want fast flash sync without breaking the bank, premium compacts are worth a look. With a leaf shutter, most of them can sync with strobes all the way to 1/2000s-1/4000s. Battery life will take a hit though.
