Start with short game activities on the chipping/putting green. Use your current ball in the test as well. You may want to create a spreadsheet to keep track of the data. Pick a quiet day on the course or late in the day. You will want to hit all the shots you would typically hit on the course - everything from putts, wedges, irons, and metal woods. You can try to blind test yourself like I did. Place the balls in a black bag and pick one out at random and then hit. Or have a buddy help you set up balls at random for you. Keep track of things like feel, distance, sound, ball action, control, ball flight, workability, and so forth.
It could be quite a bit but it is oh so enjoyable and rewarding! Let me know if you have any thoughts!

In the past I have just hit a handful of shots with both balls. You do not need to play the entire round hitting 2 shots. Hit several drive, various iron shots and some wedge shots with both balls and you should have a pretty good idea of how each ball compares to the other.

I typically do one of two things:

Play a practice round

or

Play one ball on the front and one on the back for two rounds.

If you are not allowed to play two balls during your round, I would suggest alternating balls on different holes. I would try to ensure that you split the holes not simply by even and odd numbers, but ensure that you break up the par threes so that you can hit similar shots with different balls, and the same with par fives and par fours to ensure you get a relatively equal share of drives, fairway woods, and long / short irons to gauge the effect of spin and distance control. It is not important to hit every shot perfectly either. It is important to know how the balls react not only with great shots, but also with off center hits as well. Rough, fairway, chipping and putting are all parts of the game.
Most importantly - enjoy your time as a tester and try to take some notes along the way so you don't forget your thoughts by the time you get home.
Darius

Who says that you can't hit two shots from the same spot on the course? I'd like to speak with them. Go out and play twilight rounds (afternoon). At that time there definitely will be room for you to go and hit multiple shots from the same spot.

I notice the greatest difference between golf balls off the tee box. Hitting woods gives you a great feel for the difference in distance. Also on shots 100 yards & in give you a good feel for how soft the ball is. Changing ball each hole would work without breaking any rules.

Hi, well my first response is to go out late and do the A/B test on different holes. Hit two drives on one hole. then two approach shots on another, etc. That won't impact the course of the speed of play.

If that will upset your club, then go play another course that's not so strict.

Thanks for all the helpful comments. I've been putting the balls at home and I can't tell a difference between them. Hopefully I'll notice a difference on the course.