This is a topic that is near and dear to my heart. But let's be clear we are talking about the same issue, cold bore (fouled bore) shots, not clean bore shots. While the latter is also achievable to place a shot in the desired POI, it is beyond the scope of this discussion**.
Because I am year around hunter, mostly varmints in the summer, predators in the winter, and big game in the past, all my rifles are tuned for cold bore desired POI. This is not difficult to achieve in my experience even with my more stringent requirements of 1/2 to 5/8 moa for varmint / predator hunting.
Your requirement of a 1 1/2-inch radius (3" circle) is reasonable, and I would add, enough for big game hunting. This should be easy to achieve with a mechanically defect free rifle and scope. I would also like to compliment you on establishing an accuracy standard for your intended use and equipment. And I might add, a sensible one.
Because of the modest accuracy requirements, you do not need to make this process complicated. Choose a day that the temperatures are moderate, preferably below 80 and test three loads with 3 shot groups. Allow the barrel to cool about 5 minutes between shots or until the barrel is cool to the touch and about 10 minutes between test loads.
Once you find a promising load, retest with 5 shot groups, preferrable 3 times on different days, allowing the barrel to cool between shots. Mark the cold barrel shot for each of these groups. This should give you an idea of the cold bore capability. Unless something strange is going on, they should be within the radius you specified.
I do a lot of practical range shooting throughout the year, and I have a very definite idea of both my cold and clean bore shot capability. In addition, all my range practice is done from clean and cold (fouled) bores.
** I do not need to fire fouling shots for either cold (fouled) or clean bores. My rifles are ready to go into the field whenever the opportunity is available. I have found that the clean bore issue is primarily related cleaning but as I said, that is beyond the scope of this post.