All of the above are fantastic solutions to your lining or padding problem, and all will work.
The most common and simplest being the leather liner or spider, similar in shape and function to Sir Brians liner although much cruder.
As James has suggested, the great helm would always have been worn over, at least, a maille coif and more often than not a cervellier (secret or small skull cap) as well.
Earlier great helms, and flat topped pot helms, were worn over a mail coif that was fitted over a flat topped arming or padded coif.
I think, if worn under a maille coif, the coif and ring from Steel Mastery, once the ring was adjusted to fit your helm and attached to the coif, would give the right, flat topped effect.
If you intend to use the helm for full contact fighting then I would deffinitely go with fitted, high density, soft foam (such as pipe lagging or refrigeration insulating material) shaped and fitted in such a way that it feels like a motorcycle helmet when it`s on. To draw the eye away from the modern materials, the foam should be covered with soft linnen or cotton. Thus giving a much more comfortable surface against your face and head.
As with most armour items there will always be a certain amount of tinkering and tailoring to make them work well. You will also find you try a few different solutions before finding the one that is suitable or comfortable for you.
Good luck.
G.