The pressure isn't bad due to overloading the pressure switch contacts (assuming they are actually switching the pump - if they are switching a relay/contactor that's switching the pump, they may be very lightly loaded, but that is not the usual way. Depends what's in the control box.) If the pump is 3-phase (which is possible but not the most likely at that HP) it might well be using a contactor.
The contacts on the pressure switch will tend to burn out early and provide no pressure at all (or weld closed and not stop running) due to being undersized, but that's not your main issue causing "bad pressure" right now.
That's a lot of pump. Your comment cleared up that you in fact have 120 cows to water. Your comment also stated that the pressure was fine when the pump crew left. So, the system as installed was capable of doing the job, and at some unknown point in time after the pump crew left, something changed.
If this wasn't a one-man outfit and you didn't get the head (or most experienced) person on the job, you might request a visit from the head (or most experienced) person to see what your $12,000+ isn't doing, and figure out why before you add a holding tank and another pump to a system that may not need it.
If you shut the pump off for a while, and it can build pressure when switched back on after you do that, you might have insufficient water coming into the well. Not the option I'd reach for first, since you mention "were told the pump was bad" not "increased the size of the herd" as your reason for the change, so I'd expect similar daily water use even if the new pump could pump faster (though the specifics of pump curves get a lot more subtle than the size of the motor attached. Similar GPM at more pressure, more GPM at same pressure, more GPM at less pressure are all possible when comparing specific 2HP and 3 HP pumps.) So, if the new pump is set deeper than the old one was, it might need more power to deliver the same pressure at full drawdown, for instance.
If you shut it off for a while and the pressure is pretty much the same when you turn it back on, there might be a leak in the well piping - something cracked or otherwise failed. Perhaps your pitless adapter, if applicable, is not seated correctly, or had an o-ring blow out. If you listen at the well-head you may be able to hear it (if it's actually the in-well piping leaking) - so (one option) your pump would be pumping valiantly, and much of the water would be falling back down the well. Very little pressure would make it to your end point due to it simply leaking away instead.
Another option is that it's the out-of-well pipe leaking, in which case a wet spot might be expected along the route of the water line (and you might be pumping your well dry, too.) In that case you probably would not hear anything at the wellhead, and you might see pressure build a bit after shutting things down (if it's low due to pumping water into the leak, and the leak is not back into the well.) If you build whatever pressure you can build with the water shut off past the pressure tank, and then shut off the pump, pressure should hold steady. If it drops instead, that's a sign that something is leaking, though this can be complicated by the relative locations of any check valves and the pressure gauge(s)
