Years ago there were a number of 16 & 18' caravans that were twin axle, these days it is rare to see any new caravans that are twin axle at less than 20' they are virtually all single axle, similarly, almost everything 20' and above is twin axle so it is more to do with what size caravan you want rather than one or two axles,
A 1400 kg single axle caravan will have a heavy duty axle and heavy reinforced tyres, wheras twin axles use lighter axles and conventional tyres so the weight difference is less than you'd think.
Our freind's single axle caravan has a miro of 1254kg and a mtplm of 1400 so has a very small payload allowance. Our twin axle is three foot longer with a miro of 1266kg and a mtplm of 1600 so our payload is nearly 200kg more.
Manouvering T/A's with the car is no more difficult but you are right, they are a pig to move by hand!.
Has anyone had experience of sites turning away T/A's recently?? I haven't.
If your single axle van is satisfactory but unstable to tow, you should address the stability problem first.