How can I reproduce the functionality of these 3-Arm pergola corner brackets for 4x4 wood posts? My concern is that it will need to securely hold the weight of the treated 4x4.
You'd reproduce that function with knee braces, which is what nearly everyone else does. Such hardware would need to be very robust to do the job on a larger pergola, making it cost-prohibitive (and somewhat ugly). You can find heavy-duty pergola hardware kits online, though they won't be exactly what you've shown.
However, three simple L-brackets on each corner (two on the faces and one on top), properly sized and with adequate bolts, would probably do. You'd need at least four bolts through each bracket, for a total of 12 at each corner. You can see how this gets messy in a hurry.
To rectify your apparent misconception, the challenge isn't carrying the weight of your beam. It's dealing with the racking forces applied to the corner joints. There can be a ton of torque on those points in the wind.