I personally have had a 10 gallon with a home made divider in it, but it was made of plastic mesh which had holes and the bettas could see through it (although the holes were too small for them to nip each other). The bettas grew tired of flaring at each other after the first two days and then went on about their normal business - although I believe their new form of challenging each other was to build their bubblenests right up against the partition, they never did grow out of that. Just provide them with plenty of cover and decorations so they can go sit out of the other bettas sight for a while if they want, and they should be fine.
As for a solid , no holes, no see-through material to make a divider out of? I suppose perhaps you could try finding one of those hard plastic cutting boards like you'd have in a kitchen, and then taking a power saw to one to cut it down to the right dimensions. I can't think of much else that would really be appropriate that also isn't see through. Plexiglass is see through, so is normal glass, which are the two things I most commonly hear of being used when solid partitions for tanks are made.
I really have no idea how good a betta's sense of smell is, I've never found any material about it. I mean, they have noses, or at least nostrils, so I would assume there is at least a rudimentary smelling sense, but no idea if they can smell out the pheramones of another male betta in neightboring water. I don't even know if bettas release pheramones really. I kind of always figured they were much more visual-oriented fish, being as how they make such a big fuss over displaying fins and posturing.