3k trailer is only 300# of tongue weight.
If a passenger tire can't take an extra 300# of weight on the vehicle it was installed on, you would have the wrong tire.
The main difference between a "P" and "LT" is actually in the overload protection over what the tire is rated for. "P" series tires should never be loaded to the capacity on the side of the tire while "LT" tires are built with and extra 20% margin of safety of over load built in.
Never exceed the GVR of the vehicle or risk suspension damage. Never exceed the GCVW or you risk hitch, transmission or drive line damage.
As long as all these guide lines are met, you are in good shape....happy trailering....