That's a very specific paper requirement but if you have the others on hand why not try a sample and see how it works?
Paper Temptress do carry Bright White Matte Kote (
here) - they describe it as a glossy cardstock with a matt surface for those who want the properties of gloss without the glossy effect. So if you have a glossy cardstock there and the glossy finish won't effect the finished result of whatever you're trying to achieve then that's worth a try, I would think.
As Vics says, Distress ink is indeed dye ink so yo should be fine with that.
Let us know how you get on!