today in our other car, an AUIII Falcon, I drove onto a small patch of mud and got bogged. Only one back wheel was on the mud, the other on a dry and sound surface, but zero power went to that wheel, it all went to the one that spun. Presumably if I had an LSD it would have been fine: drive to both wheels.

My Q is: how would my FWD TW have performed in this situation: can FWD send all the drive to the one wheel without traction?

There is an LSD available, but it is not standard. Only in the Ralliart manuals.

Exactly the same, the only slight advantage MAY have been some more weight over the drive wheels but in general you would most likely have still been bogged

It's also worth noting that not all LSDs will help you in this situation, or will to varying degrees.

Helical LSDs which myself and a few other members have fitted in the form of a Quaife LSD - and the Ralliart LSD - are a "torque multiplier" type LSD, the problem is if you literally have 1 wheel in the air, the remaining wheel will still do nothing, because 0 torque is applied to the wheel in the air, so any multiplication of 0 is still 0 to the other wheel. However you can help some torque onto the wheels by using the brakes - A hummer for example has this in the manual - use brakes to get un-stuck in mud/snow.

Obviously in the mud you have a little bit of torque but if its really really slippery it might not be enough.