As a RV'er since '94, have seen more overweight, mis-matched tow rigs and trailers than you can count.
Something you need to consider is the wind load. The front of a travel trailer is like putting a piece of 4 x 8 plywood sheet on the top of your car and trying to drive 70. Another factor to consider is a non-turbo engine looses about 3-3.5% of it's torque per 1,000 ft of elevation change. so go out west, climb Monarch Pass, 11.300' at the top and you will be thinking what the heck am I thinking.
The other issue is going down the hill. With a 4-5K trailer you will need a trailer brake controller for the trailer. The engine will not have much retarding going down, so ride the brakes. A sure way to need the run away truck ramps if you make it that far.
You might get a class 3 hitch, 7 pin wiring plug, a brake control, equalizer hitch and a aux transmission cooler. Go out west, and when you get back go buy a truck.
Do the math, LP weighs 4.25 lbs per gallon, water is 8.5 lbs per gallon, add a battery or 2, look at the GVWR of the Pilot, on the door jam, go weigh the Pilot as you will be using it, a couple of folks, a cooler, and gear. now subtract that weight from the GVWR, of the Pilot Probably close to max. Plus the weight of the hitch and equalizer that will be on the car.
The towing 8K with a Pilot, shear STUPID, irresponsible to the other drivers on the road and a good atty would own everything he has if in a serious wreck.
Bottom line, you would probably be fine towing that trailer to the lake 3 or 4 times a year. Cross country, not so much.
My tow rig: About 27,500 lbs total.