If you are even going to bother wearing a helmet you shouldn't put a beanie under it. The reason being that it makes it bigger. The reason you don't want it to be bigger is that when you smash and hit your head, it allows the helmet to move and there for making it less effective. Which is why they make helmet liners that are very thin but add a bit of warmth.
If you want the look of a beanie underneath, go with bern. It comes like that. However, You need to make sure you are buying a helmet and not a hardhat.
Hardhats are meant for keeping things from going through your head. Like if you smash your head on a rail, you wont crack your head open, but in no way will it keep you from getting a concussion.
A helmet will keep you from cracking your head and help ease a concussion but most likely wont prevent one. I believe a helmet only has to be effective in a 1.5 drop and only at around 15mph in the US cause we don't regulate that well... so in most cases, your helmet really doesn't help much. It just makes it so that if you slam your head, it wont split open (hopefully) and your concussion might not be as bad.
Also, one major fall warrants a new helmet. So if you are spending more than $100 you are spending too much on a one time item.
(I sell helmets and goggles for a living)