Leave your tire pressures alone for purposes of this discussion. Tires are just one element. There is also the wheel rim rating, the axle rating, and the spring rating. Your axle ratings are based on all of those, and consider the recommended tire pressure.
Your two axle ratings don't add up to your gross vehicle rating, by design. That is because the vehicle designers didn't know where you were going to put your payload, more to the front of the bed (loading the front axle more) or more to the rear (loading the rear axle more). Your GVWR is still your GVWR.
If you are not a commercial carrier and subject to federal laws about GVWR, you can decide to load your vehicle over the rated GVWR, but not exceeding the axle rating for safety. Some will disagree strongly with that, and any time you carry more load you need to be aware of the need to drive with consideration of your load and conditions, but GVWR is not a magic number. The manufacturers will put a smaller GVWR sticker on many trucks just to get under the tax rules related to GVWR, and not change the vehicle. If you decide to exceed the GVWR that is your decision. I wouldn't recommend you exceed the axle or tire ratings under any circumstances.