I play a lot of backgammon, and I think this is a great idea. The doubling cube is my favorite part of backgammon; it's use is 100% skill, and passing over the cube can really give your opponent a headache. As others have pointed out, to make it work you would need to play for money or play a match to a certain number of points, and the latter sounds preferable. Playing a match may not be idea for online play, but should work well IRL. When I used to regularly play backgammon with my dad, we would play to 11 points, and just tally score over how many evenings it took to finish he match.
Having the ability to score two or three points is a big part of what makes the cube interesting, so adding that possibility to the dominion game would be a nice addition. My intuition is that something like this would work: If you win by 12VP or more, then you get 2 points, and if you win by 36VP or more, you get 3 points. These numbers correspond to a clean 5-3 and 7-1 province split, respectively. In backgammon, scoring a double game is difficult but doable, while scoring a triple game almost never happens, and requires quite a bit of luck.
As for doubling cube strategy, you generally want to double when you have a 75% chance of winning. At this point, it a wash for you opponent as whether or not to accept the double or drop. If they accepted during 4 different games, they would win one game for 2 pints and loose three games for 6 points, a net loss of 4 points; while if they dropped during each of these games, they would also loose 4 points total. If you double earlier, you unnecessarily risk the chance of the odds going back in you opponents favor, while if you double later, you will loose points by your opponent dropping. Of course, this often isn't so cut and dry. For example, if you were building a megaturn deck in dominion, you'd likely need to double before the megaturn hits, as doubling afterword would result in a clear drop for your opponent.
And to make it somewhat parallel to backgammon, I would say doubling before playing any cards on your turn would be logical.