That trailer is definitely too much for the Explorer before you leave the dealer's lot. Maximum towing weights in the vehicle towing guides are not realistic. How much a truck can pull is individual to that truck. The tow guides are stated for the lowest trim line with minimal options, a 150 lb driver and minimal tongue weight. Think a four wheel farm trailer where the weight is carried by the trailer. The same goes for trailer dry weights. A trailer will weigh more before you even leave the lot once the battery and LP tanks are installed. Commonly the trailer will be heavier by a 1000 lbs or more. Most dealers won't tell you this as they only want to see the trailer leave the lot.
The first thing most tow vehicles run out of first is payload capacity. The TV will have a sticker stating the maximum payload for the vehicle on the driver's side door pillar. This is the maximum weight that can be loaded and includes the weight of all passengers, their personal effect, cargo, owner added options, weight distributing hitch and trailer tongue weight. To determine the available payload for the trailer tongue and subtract the weight of all passengers, personal effects, cargo and 100 lbs for the WDH. The result is how much weight you have left to carried the trailer tongue. Take this number and divide by .13 (13%) and you will have a good approximation of the maximum trailer weight the TV can tow safely. Tongue weights typically run between 10% - 15% of trailer weight. The most common range is 12% - 14% which is why I use 13% as a rule of thumb. I would also calculate for 15% tongue weight to obtain a worst case scenario. For your trailer assuming a not unrealistic a loaded weight of 6K lbs you would have a tongue weight of 780 lbs. Don't forget that kids grow so that 40 lb future linebacker in the back seat could be closer tot 200 lbs down the road.
After the payload you still need to determine if you are within the Gross Combined Vehicle Weight, hitch receiver capacity and Gross Axle Weight Rating. If any is exceeded then the TV is too small for the TT. The GCVW is obtained from the towing guide or owners manual. The hitch receiver should have a sticker with its ratings and the GAWR is on a sticker on the driver's side door pillar. With the exception of hitch receiver rating these are most accurate by weighing. You can get a good enough quick estimate the GCVW by adding the TV curb weight plus estimated passenger/cargo/WDH/tongue/etc weights plus the estimated loaded weight if the trailer. Curb weight is obtained by taking the Gross Vehicle weight Rating from the driver's side sticker and subtracting the payload from sticker. I have yet to see the GAWR be exceeded by a TT that meets the payload, hitch receiver and GCVW. I think it is more of an issue when you start talking 5th wheel and goose neck trailers where the pin weight sits right over the axle. I would not recommend towing a 5th wheel with a 1/2 ton truck. There are some that can handle the 5th wheel weights but a 3/4 ton also has the truck weight to help keep things under control.
All of this should also be in the owners manual's towing section.
With that said there would be a number of F-150 configurations that should tow that trailer just fine. Personally I would go with a 3.5L Ecoboost but the V8 would also handle it. You should look at XL, XLT or Lariat trims with nothing more than an x02A options package to get the highest payloads. While the math might work for a 2.7L model having one I would not recommend that for this trailer. While capable to me the practical maximum is more in the 5K - 5.5K lbs range. You can get a better idea or get questions answered on the F150 forum. In particular the towing section. I would also spend some time browsing the "Post Your Payload" thread to get an idea of the payload ranges you will run across. Here is a link to the towing section:
https://www.f150forum.com/f82/