Hoo, boy.....this topic can be a very complex can of worms. For CERTIFICATED aircraft, there are many factors that can dictate (or limit) maximum gross weight. As others have noted, climb rate and load factors are high on that list, but consider that, for example, engine cooling can limit climb rate.....not because the airplane can't climb at a higher rate, but rather, the engine cannot remain within required temperature limits while sustaining that minimum climb rate. And, bear in mind that this testing has to be done as corrected to maximum operating temps.....
Also, landing gear drop tests are required at maximum landing weight, which, for light aircraft, is typically the same as max takeoff weight, so that can limit GW.
Etc.
When it comes to experimental aircraft, the maximum gross weight is whatever the builder sets it at. And, that may or may not have been based on empirical testing, or any testing at all. I assume that all the kit manufacturers do some at least empirical testing, if not genuine stress testing on their airplanes, but if a builder buys one of those kits, he is not bound by the recommendations of the kit manufacturer.
To the OP: If your mission is indeed four people and gear, on amphibious floats, the MINIMUM airplane I'd be looking at would be a Continental powered Cessna 206F or G. Even the 206 on amphibious floats can be very limited in useful load. It is possible to lighten the airframe up some by reducing the weight of the interior "decor" massively, and some other ways, but amphibious floats are HEAVY.
The Murphy Moose on amphibious floats in my limited experience (a few hours in a Vedenyev powered version) doesn't have enough vertical fin and rudder for the float application. Performance was great on water, but we weren't heavily loaded.
In any case, what you are asking for (your mission profile) is a LOT for an amphibious airplane. And, if you also need good takeoff performance, think deHavilland Beaver rather than Cessna. At that loading you are describing (and remember that FUEL is also a fundamental part of your useful load), ANY Cessna amphib is going to be barely able to carry that kind of load, perform well and go much distance. In your description of your mission profile, you didn't describe what sort of range you would be considering. Remember that, by definition, amphibious floated airplanes are pretty slow, and because of drag and weight, they burn a lot more gas typically than their wheel equipped brethren. So, don't forget fuel loads.
Bigger engines offer better takeoff performance, which is vital on floats, but they also burn more fuel. Granted, if you have the discipline to pull the power way back on that big engine in cruise, you can burn the same or close to the same fuel per hour as the smaller engine......if you have the discipline. I rarely demonstrate that kind of self control.....
I don't know what the status of Murphy Aircraft is. The Moose seems like a pretty good airplane. I would stay away from the Vendenyev engines for the most part, for a lot of reasons. But, they are much cheaper than a good Lycoming......and they make power. If you are willing to tolerate some of the "unique features" of those Russian engines, they do work.
But, I'd be doing a LOT of research before I pulled the trigger on any one aircraft type.
MTV