Bad idea, the pump does not suck very well at all, it pushes water. While an OB can placed in a tub works well because the pump is submerged, the I/O needs to have water half way up the transom assembly to do the same thing. My guess is your new pump impeller needs replacing

AllDodge-I’ve never tried it but always assumed if pick up holes were submerged it was fine. Does the higher level when in the lake actually push the water up into the pump?

This does not make any sense to me, maybe I do not have a good understanding of the system. Are you saying the motor does not intake water through the pickup holes on the lower part of the out drive?

Answering this, but not on your overheating issue....
The rivets in the tstat housing gasket ground it to the intake manifold. The sensor is grounded to the tstat housing.
So, also make sure both gasket surfaces (housing amd intake manifold gasket surfaces) are not painted .

Yes it does, but (always a but) the pump has a very hard time sucking (as in creating suction) when there is no water in the vanes of the pump. The pump is above the drive port holes, this is why muffs are used and not a tub. Using a tub on an Alpha drive can be done but not a belt driven Bravo

In an OB the pump is just a short distance above the intake holes and placing the leg in a tub will submerge the pump.

You need to put the boat in the water and try it. You are most likely chasing a problem that doesn't exist.

You think my overheating problem doesn’t exist? Like it will get hot on the hose but not in the water?

Correct. A hose will not supply near as much water as the pump will pump. It will suck a garden hose flat that is turned wide open with 80# water pressure. Drawing out of your bucket doesn't work because the pump is trying to lift the water 2-3'. With the boat in the water, water level is almost level with the pump and it only has to lift water a couple of inches. When underway, water is forced in the pickups and water pressure is higher that the pump alone can produce. I could be wrong, but I suspect you are chasing a ghost and you may have already damaged your new impeller.

If it has a pump with a plastic housing, change the housing when you change impellers. Buy the whole kit from Mercury. Don't try another brand, don't just change the impeller, It rarely works out.

Clarifying thought
If a Bravo motor is run on a hose using ear Muffs, and stay below 1200 or so rpms it should not overheat
If the same motor is Reved to high, the impeller can go dry and damage the impeller

If a Bravo motor is run in a Tub of any kind, the motor can overheat, It can also damage the impeller in the pump while doing it

If the pump and impeller are currently have not been damaged, then putting the boat in the water it should not overheat "unless" there is another problem that has yet to be determined