I'd have a look on some forestry/arb suppliers.
They say u can use blue poy on those winches but need an extra wrap or 2 to get the right friction, 10mm + b;ue poly has a decent breaking strain considering ur only pulling something 200kgs max
Various types of 'Bull' rope or any of the lowering or rigging rope, all designed to be worked in winches and lowering trees down so minimal strech and fairly burn proof. (Try likes of Northern arb supplies, Clark forestry, honey bros, fr jones, buxtons, gusharts are a few of top of head)
Also tree climbing/arb rope is different to rock climbing rope so has no strech in it, so if u know any tree surgeons meant to be flung out every 5 yrs so they might have old stuff lying about but many repurpose it for other jobs.
200m of rope will take a fair bit of managing, the only way u'll hope to do it is have a drum/bag it fits in and just feed it in the drum. (u'll see rope bags on arb sites)
If it was me i'd be working 2/3 ropes and joining them, so easier if u get more shorter distances.
Various ways to join possibly simpiest 2 bowlines so will always undo, just mind to leave a big tail on the bottom rope so when untied has enough to wrap round the winch.
Another idea might be to make a 'winching cone' basically a strong traffic cone u put over the deers head so t should not catch objects/hanks the same and just slides past. Some folk are using them for logging so must work to some extent.