A 4 cylinder engine can handle a popup of a couple thousand lbs ok, but there's more to towing than that. You want adequate engine and transmission cooling, a transmission that isn't marginal, good brakes, a receiver attachment that is strong enough to not tear in actual use, and adequate rear suspension.
Some people tow popups of that size with a Subaru. They have a couple of models with 2700 lb tow rating, but only 200 lb allowed hitch weight. This will force you to be careful with how you load the popup with cargo so as not to exceed this limitation. But it can be done, and is done. If you tow a whole lot or tow with excessive speed (over 60 mph, but slower up grades), the life of the drive train components could be compromised.
I've been there and done that. I towed a 13' UHaul 'egg' travel trailer 2000 miles on a vacation back in the '80s, using a 1984 Dodge Omni 2.2L 4 cyl with stick shift. Under 100 hp, so we were driving 50 mph in 3rd gear much of the time. Did we have a fun time? Yes. Would I do that again? No.
You will be more satisfied in the long run if you have some excess capability in the tow vehicle, as opposed to running on the ragged edge. Some good vehicles to consider (assuming you are looking at new models) would be Ford Escape with 2L turbo, Jeep Cherokee V6, Hyundai Santa Fe V6, Honda Odyssey, Dodge Grand Caravan. As kvangil said, anything with a tow rating of 3500 lb or more should be good.
My Highlander hardly knows that my 1400 lb (dry) popup is behind it.
Mike G.
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photo: Yosemite Valley view from Taft Point