Assuming you want to keep everything within the legal weight limits, it can get a little complicated and require a trip to a scale.
First thing you would need to do is put everything and everyone in the truck that you would normally take with you when towing. This includes a full tank of fuel, the hitch, etc. Weight the truck and see what you get - this gives you the gross weight of the tow vehicle. Subtract this weight from the GVWR of the truck to determine the maximum pin weight for the trailer. For instance, let's say the scale says that everything weighs 7,000 lbs - subtract 7,000 from the 8,800 lb GVWR and you have 1,800 pounds remaining before you exceed the truck's GVWR.
Pin weight for a 5th wheel is normally around 20% of the entire trailer weight, so if you had 1,700 lbs available on the truck, your maximum allowable trailer weight would be around 8,500 lbs. That's assuming that the trailer is correctly loaded so that 20% of the trailer weight is on the pin and 80% of the trailer weight is on the trailer's axles. The entire weight of the trailer plus the entire weight of the truck will give you the gross combined weight, and this must be no more than the truck's gross combined weight rating.
The factory numbers are always high compared to what you can really tow while remaining within legal limits. For example, Ford may say that your truck can tow a 16,000 lb fifth wheel. 20% of that is pin weight, so you would be looking at 3,200 lbs of pin weight. Subtract that pin weight from your trucks' GVWR of 8,800 lbs and that would mean that your truck with everything in it cannot weigh more than 5,600 lbs in order to stay within the truck's GVWR. Chances are your truck is going to weigh a lot more than that. My 2020 F-350 has a GVWR of 11,500 lbs and my 14,500 lb fifth wheel puts it pretty much right at the maximum legal limit, even though Ford says the truck has a 5th wheel towing capacity of almost 22,000 lbs. I would be about 1,600 lbs over my truck's GVWR if I towed a 5th wheel that weighed that much. The problem is that the vehicle manufacturers say that the tow rating is for a truck that is "properly equipped", and by that they mean a base model regular cab 2WD truck with practically no options. Every extra pound added to the truck will cut in to your maximum tow rating, so a fully loaded 4WD crew cab cannot legally tow as much as a lower optioned model with the same power train.
Switching gears from 3.55 to 3.73 will make it easier to tow the trailer, but it will not increase your truck's legal towing capacity. That is set before the truck leaves the factory and cannot be changed regardless of what you do to the truck. For legal purposes, the sticker on the door with that has the weight ratings is what matters. If you get into an accident while towing overweight, there's a good chance that it will be identified during the investigation and this would certainly not work out in your favor and possibly even make you liable for an accident that might have otherwise not been your fault.
Many people are very surprised when they run the numbers and find out that what they can legally tow is far less than what the towing brochure says.