Lazyplayer suggested here a method that could allow for drawn games. I think his main motivation was to preserve symmetry, but symmetry implies the possibility of draws I guess. It is also supposed to reduce the importance of memorizing an "opening book".
Here's a modified version of that proposal:
1. Prepare a list of all the N move opening sequences that lead to an approximately equal game. (Perhaps N=2 or 3?)
2. Pick a opening from this list at random and present it to the two players.
3. The two players pick their preferred color/side independently.
4. The players reveal and compare their choices. If they are the same, then go back to step 2. If they are different, the game continues from the opening with each player playing their desired side.
5. If step 2 gets repeated M times (e.g. M=5)---in other words if the players agree on opening after opening---then the "game" is declared drawn.
Obviously this method has the downsides that it would require some effort to build the list from step 1 (presumably using the strongest available bot). It also requires some way to randomly sample, as well as make choices in secret. OTOH the gameplay would still basically be Hex, the situation is totally symmetric, and memorization would be impractical.