I'm not an expert by any means, nor do I portray a Templar persona.

But, I tend to be intrigued by research sometimes. Looking at some of the literature to hand, it seems elusive to try to pin down a singular cloak style.
Certainly, fighting in the Holy Land, a knight would likely acquire clothing of a local "flavor". I would assume a burnoose would be passable.
The few illustrations and guidelines I could dredge up leave a lot to be desired;

Unless you are a knight, I would reccommend against white for a cloak. (Besides the obvious cleaning issues.)
Black or brown for a sergeant or man at arms, with a 9" square red cross formee' on the left shoulder. Or, if your personal is a priest or chaplain, green appears to be the color choice. Just no fur edging, and made pref. of wool for all cloaks. Most appear to be mid calf length.