If it's important to know which day a golfer played, you could...
Signup day 1 players in round 1 and day 2 players in round 2. Only post the results to the round they played in and print out results accordingly. You are only using "rounds" to indicate what day a golfer played. A golfer will only have one score posted, either in round 1 or 2...so the total for any golfer across the 2 rounds will only be one score.
I would create a custom field (select OPTIONS on the SETUP menu)
called "day played" and mark the golfer with either day 1 or day 2.
Create a "custom" view that includes the "day played"...you can use this
Tip: To quickly put golfers in day 1 or day 2...., highlight the players you wish to change the value for. Click the [Edit] button. Type in the appropriate day value and click OK. All players will now have this day value.