Charlotte here...You're best off with sticking with lightweight wool instead of linen. I've worn both in hot summer months, and not found the wool to be any warmer than the linen. In fact, if you get a sturdy linen, the linen can be quite warm.
The most important thing is to have a layer of linen next to your skin, as your shirt. This will keep you cooler and comfortable.
Historically speaking, wool is almost always the better choice for your non-underwear clothing.
As far as your stature, a gown would be good idea. Shorter, no fur, lighter wool will keep it cooler. Also go for a lighter color. Most of the illustrations show people in only their doublets when they are actually working, so wearing a gown the rest of the time should be appropriate.
For your archer portrayal, I'd defer more to the military folks, but I think you'd want some sort of arming doublet or padded jack. I believe that it's appropriate for these military garments to be linen (though a jack can be many many layers, worn over another doublet, and wouldn't be cool at all!)
Cheers,
Charlotte
(Not Jeff at all - just married to him!)
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Geoffrey Bourrette
Man At Arms