Current United States
Would it be possible for congress to enact two sets of laws for healthcare, one authored by Democrats and the other by Republicans? What I'm suggesting is a single, large bill that starts with a clause stipulating that the president may choose which half goes into effect.
Possible? Sure. Congress has delegated many such powers to the president. Likely? No.
The problem is that the president is not separate from the party system. If that had happened during the Barack Obama administration, people would have known that only the Democrat version mattered. Now, it would be surprising if anything other than the Republican version mattered. And that was even more true during the George W. Bush administration. So Democrats won't regard changes to their half of the bill as productive, leaving them unwilling to help with the Republican portion.
Even if we had an independent president, this still wouldn't necessarily work. Consider what happens if Democrats and conservatives work together. They can make the Republican side more conservative and less likely to be chosen by the president. Then Republicans and liberals can work together to make the Democratic side more liberal for the same reason. So rather than have a somewhat stable compromise, one of two extreme measures would pass.
Don't think that they'd be that self-serving? In 2012, Claire McCaskill of Missouri ran ads depicting Todd Akin as the real conservative among Republican candidates, helping him win the primary. Then she narrowly won the general election. Of course, the same strategy didn't work nearly as well for Hillary Clinton.
You seem to be trying to to develop an equivalent of the cake-cutting solution. Note that the cake cutting problem solution of having one side divide and the other side choose who gets which division works because size is the only criterion. Because both sides agree on that, the cutter divides the cake as evenly as possible under the assumption that the cutter will get the lesser piece.
Here you don't care as much how big the slice is as which side. Here, the presumption is that the president will pick the bigger slice. The goal is to get as much of your platform in it as possible. So the incentive is to fill your side with as many popular features as possible--coherent or not, so that the president picks your side.
You can actually see something like this in referendums. Often, there is a status quo choice where the situation stays the same with no changes and a change choice. But it is possible for the referendum to be picking between two changes. For example, the current Puerto Rico referendum is choosing between statehood and increased independence. There is no status quo option. The two sides work together to define their own positions and leave it to the populace to choose.
Notice that statehood is a more precise position than "independence". This is because previous referendums have produced no majority winning position. So they keep reducing the choices so as to encourage a solution. The last time, the referendum was clearly against the status quo. Of course, that combined statehood and independence, even though those are not a single coherent alternative to the status quo. Thereby they defeated the status quo and the status quo votes have to choose between statehood and increased independence.
Statehood is probably closer to the status quo than full independence. But independence is two options: free association and independence. So someone who prefers the status quo to one of the independence options but not the other may vote independence in the hope that they'll win the future fights over defining independence. Who better understands voters? We'll have to see how it turns out.