This is not at all helpful, but I found my perfect linen tee at BR last summer. I am praying they bring it out again, because I discovered it was so perfect after most colours had sold out. I bought two black ones, and if it shows up again, I'll stock up. I seem to recall they had it in white, which I found thin, but the black isn't at all. So this is just a heads up to keep your eye on them, just in case. Find below.
I was on a bit of a linen tee mission last year, had a pretty good success rate finding ones with free or minimal shipping cost to Canada...Zara, BR. I also got a Madewell one from Nordies. Keep your eyes open, and try anything you can. And post if you find something awesome