Duel:
Let's look at this in terms of the strongest cards, bar none, for each of your deck types.
Suicide Black: Cards like Sarcomancy, Carnophage, Black Knight, Fallen Askari, Dauthi xxxx, Unholy Strength, Paralyze, Erg Raiders.
Stompy: Cards like Rogue Elephant, River Boa, Ghazban Ogre, Wild Dogs, Elvish Archer, Giant Growth, Harvest Wurm, Spectral Bears, Symbiosis.
Given these cards, I think the matchups should be basically even, with the edge going to whoever gets the most creatures and enhancers early on.
Like the others, I don't know what your brother is using in a stompy deck that's so expensive, as a no-holds-barred stompy deck shouldn't even need Gaea's Cradle (unless you're packing Overrun).
Anyway, I think the tendency here is to give the nod to the cheap old cards. If you look at some of the cards that came out in earlier sets that are no longer legal in Type II (Black Knight, Rogue Elephant, Unholy Strength), they tend to be stronger than their current counterparts (compare White Knight vs. Longbow Archer, for instance).
To answer your question, though, I think that if you played Type II-legal suicide black, you'd probably lose to a Type II-legal stompy deck, but if you played a black no-holds-barred deck made up of powerful rares like Mox Jet and Juzam Djinn and Nether Void and Underworld Dreams, you should blow away any Type II-legal stompy deck that for some reason cost the same amount of money as your no-holds-barred deck.