Will replacing my jet pump with one that has a higher shut-off pressure increase the discharge pressure?
With 15’ of lift you would loose about 7psi of pressure “static” I would be looking at your pressure switch, your system should be able to pressurize to ~62 psi with the long lines to the lake I would expect there to be large drops while water is being used but the tank pressure when full should be much higher, it is possible your “jet” is worn or your pressure switch needs adjusting I would look at these before just replacing the pump.

Added to address the edit. Since you can not get a higher pressure “static” now it is down to just a couple of possibilities, the jet is failing these are a fancy Venturi that creates the ability to push water it could be worn out (it happens) the pump may not be developing the needed pressure because of wear. I would want to verify the output pressure of the pump, if it is good then replace the jet (not that hard and less expensive than a pump. These are just about all that it can be (since you did not say anything about leaks).