Polyester is fine as a synthetic base layer. The thing I don't love about it is that it tends to trap bacteria from sweat and ends up stinking! - even after a normal wash. Adding white vinegar to the rinse cycle helps break it down and neutralize the odors however.
If your mtn biking ones are thin, you'll probably want to layer more over those OR consider a thicker base layer. I generally use my thinner base layers in spring conditions.
This link has info on base layer tech and materials Underwear (Base Layer): How to Choose
Bamboo (which I don't think is listed on that link) is a natural fiber and renewable resource with wicking and antibacterial properties. I have bamboo socks I use for another sport and find the wicking is good and they don't smell like they trap bacteria from sweat. Also have a couple of thin base layers in bamboo that I haven't had a chance to test in riding conditions (spring).
I stay away from wool because imo, wool is intended for sheep
But it does offer warmth and wicking, and people tend to really like it as a base.
Using that Burton jacket you linked might be overkill as a mid layer under your new shell. I'd say it all depends on if the combo is too heavy and/or restrictive. If you're not in super heavy white out conditions or spending a lot of time with your upper body on the snow, you might find that jacket is suitable as your outer layer. Just throw a base and mid layer beneath it. The weight (warmth rating) of those under layers will be based on how warm or cold you run. For example, you might be fine with the thin base and a light to medium mid layer under whatever thermacore insulation means. Or you might need thicker base and mid layers. You'll have to see.
If it turns out the Burton jacket doesn't work under your shell, here are some links to give you an idea for a mid layer. In a mid layer, you really don't need waterproofing, since you'll have your shell for that. But you WILL need breathability, which is what fabrics like fleece do.
Men's Fleece Jackets | evo
Men's ColdGear Clothing, Jackets & Accessories - Under Armour: ArmourŪ Fleece | US
This link has a little of both, but mostly base layers to give you some options if you decide the thin ones you have are not enough Men