The evidence for cotton in England between 1150 and 1300 is nonexistent. No clothing item I am aware of has been found in England, or within 250 miles of it for that period. The first recorded use of cotton in England, specifically cotton as in the now accepted context and not as in the finishing of the cloth, was in a surgical document. It was recommended as a packing for deep wounds that should be replaced daily to clear the wound. Initially it would have cost more per ounce than gold so you’d have to have been fantastically rich to afford the services of a surgeon and his expenses! Way, way beyond what a knight could have afforded – probably only in the realms of a King / prince etc.
By 1360 you have the first specific documentation of rice, another item initially used by surgeons and by 1360 actually used as a food item. It was so valuable that was mentioned in a will, along with castle, lands, hunting rights, fishing rights and silk dresses and a fine wardrobe of clothing etc. No cotton clothing though by that date even in written references
One common error is that people wear their silk clothing next to their skin – but evidence clearly indicates that a fine woollen tunic/dress was first put on and the silk garment was worn over it and thus protected from sweat and a person’s dirt etc.
I am aware of many items of cotton from the fringes of Europe by the mid/late 13th cent – have seen a few as well. But it was not an Easy Jet age! Any person who went on pilgrimage to Santiago, or Rome even as far as Jerusalem was very lucky to make it there alive and almost regarded as a living miracle to make it back alive! Don’t forget that the journey there to Jerusalem was enough to wash any person of their sins and the majority who attempted it did not make it back.
The evidence of the fist crusaders who went to the Holy land and returned to Europe shows them unchanged as there was clear documentation later on in the late 13/14th cent of them pouring scorn on those who’d stayed and adopted the clothing and styles of the areas there were in. Real soldiers spent fortunes on the best arms, armour, mounts and their households. Of interesting note are early illustrations showing the horses in tent and the soldiers sleeping rough on the ground. Go figure some priorities from that.
But in the current context why not use linen? The cost for good linen is £3-5 pm and good quality cotton is the same. Linen there is a vast wealth of information about not only of the thread counts, but also the dyes used and the different weaving patterns. There is nothing of any such for cotton....
I suppose the main question is what are you trying to be? Some globe spanning medieval soldier with an item of kit from every corner of Europe or an English soldier? For the Anarchy most English soldiers were employed and remained in services in England there were large numbers of Flemish mercs also in service here. Normans who went to Italy stayed there, or did brief journeys home to raise additional troops for their adventure in the south. The drift in many cases show knights leaving England and France in the 12/13/14th cents and very, very few coming home. Why come back to an area where you can’t buy lands or estates, from areas where there were kingdoms for the taking...