After the first trial game using the original rules, we also figured the First Player issue. We also changed the game such that every position (not just the start player position) are determined according to VPs. Ties were broken by previous turn order. It worked excellently! You just only need to present something to indicate the turn order of each player. The game is so much better, and makes better sense.
I'm thinking about something simple, a card with four positions numbered 1 through 4 in which you put one of the action pawns.
In fact, Netrunner's runner's action card fits with that idea: