Won't be of any benefit or very little, i've carried out similar experiments on my truck. Check the tyre sidewall, If the tyres are labelled as M+S (mud and snow) they are already considered an aggressive tread pattern and will be about the best you can hope for. Even aggressive tread fills up with mud quick enough.
In truth, its the differential that is killing you. As soon as one wheel breaks traction, then all the power goes through that wheel and you are sunk. A limited slip diff would help. A lockable diff would be better (don't think they exist as standard for transits).
I am forever getting stuck in my 7.5 tonne truck at work. We do a lot of muddy fields, woods, farms, unsurfaced yards etc, and as soon as one wheel loses it, that is the end. It is generally only made worse then as the more you attempt to get out, the deeper you dig.
There is a bit of technique that can help, but it's only ever rescued me 10% of the time

It's usually time for the long walk to the tractor.