You're aiming for around circa 1300, right? Revival Clothing makes a fantastic gambeson, but it's decidedly later in period. Everything changed so rapidly in the 14th century, that the pre-1300 styles began to rapidly vanish in the first quarter of that century. Unfortunately, almost nobody makes the longer styled gambesons that go with mail kits dating to around 1200-1320. I had to have mine custom made (I had Jess Finley of
Fuhlen Designs do it).
GDFB makes one that looks the part, but it's so thick and heavy, that it's clearly not meant as an "aketon" that goes under the mail. It's armor by itself.
What I do to cheat, frequently, is to use a long-sleeve linen tunic instead of a gambeson, just to save on heat insulation, and thickness/weight. Not accurate at all, and no armor value, but if it's under the mail, a lot of the time it's not that visible.
You can still cheat with the more 14th/15th century gambesons too, but they're expensive enough that you might not want to do that and chew them up with the mail.
It's a big judgment call here.