Welcome

We have many owners who have caravans here and tow, sure they will be along soon.
2.2 should be fine to tow, you will prob have to keep it in 'sport' mode most of the time to keep the revs up. A member here used to tow a fairly large van with a 2.0 Auto Omega, no probs

All Omega Auto's have built in coolers in the autobox, although it is recommended you change the oil in the box more regularly as the box is under greater stress.
As for LPG, we have that covered too, with many members here running LPG on Omega's from 2.0's 2.2's 2.6 and 3.0's. I am no expert but generally go for the SGI systems, these inject the gas individually into each cylinder, rather than a 'mixer' system, which does it one 'go'
Someone more experienced in LPG will explain that further, the mixer systems will be cheaper than SGI. For a professional install of LPG, your looking around £1,500 - £2,000, maybe less. The 2.2 has a lot of space, so might get it cheaper. Looking at around £800 if you DIY, a member here done it himself!
Good luck, and welcome again
