Having towed a similar trailer with a 1500 and a 3500, I can tell you that while you CAN tow it with a 1500, you will be at the engineering limits of the vehicle (if not over).
The advice to figure out your wet tongue weight for the trailer is key. I would guess the number will be something like this:
1. Truck payload capacity of 1600 lbs.
a. three passengers = 600 lbs.
b. gear in the bed = 200 lbs.
c. tongue weight of trailer when wet = 1000 lbs.*
2. So you are a little over the payload capacity listed on your door jamb. (Don't use any other number. The Brochure or what the web site says about your model, or what the salesman promised don't matter in the real world.) You are probably also over weight on the rear axle, though a properly set up weight distribution hitch can solve that by moving some of the rear axle weight to the front axle. Note that air bags don't solve that and can mask the visual cues that the WD hitch needs adjustment.
3. You may or may not care that you are over the specific limit. Your insurance company may or may not care. The state in which you are traveling may or may not sometimes weigh recreational vehicles.
4. But regardless of #3, your setup will be much more susceptible to driver error, inclement weather, sway from passing vehicles, poorer handling, more difficulty in rough terrain/mtns, and in general more stress on driver and passengers. (Since you are asking this question, it is reasonable to assume you are not an experienced tower, so these susceptibilities will likely apply to you. This is NOT intended as an insult or attack on your capabilities. We ALL start from the same place.)
This doesn't mean the towing can't be done. This doesn't mean people don't do it every day. This simply means it may not be advisable, or worse.
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*Don't believe the tongue weight numbers the manufacturer gives you for your trailer. They often dont include having propane tanks, batteries, etc on the tongue, water in the tank, etc. A couple of propane tanks, plus a battery or two, and some stuff in the trailer, can add as much as 400 lbs to your tongue weight. Every one of those tongue weight pounds counts against your truck payload capacity.