TL;DR: How to get around 3.0:1 LSD differential for XE10?

Hello! I have a 3UZ swapped IS200 which has the stock Sport-trim LSD differential (ratio around 4.1:1). It is a street car.
The drivetrain is working just fine, but I need a differential with lower ratio. Right now running around 3000rpm at 90km/h, which is less than ideal for 4.3L V8. 1st gear is almost useless, I often take off in 2nd gear and skip gears.

The internet is full of suggestions on "how to get a higher ratio", but I need a lower ratio. I know the easiest option is to find a manual IS250 or manual IS300 differential that should mostly just bolt on and have around 3.7:1 ratio. The issue is, it is hard to find one (especially knowing that I want LSD), also I believe 3.7:1 is not good enough for me.

So, I have been looking into IS220d differentials which are very well available and have different ratios, something around 3.2:1 being perfect for my application. I also know that their housing is bigger than would fit for XE10. But I wonder if their 208mm gearing would fit into the housing of a IS250 differential...

Or maybe there are other even better options
Thanks for any suggestions!

A video of the car for you to enjoy