Dip-Therm by Dometic AC
Excellent question on generator size. I wished the answer was as simple as the question but the answer involves lots of stuff.
First, of all lets look at the AC unit itself. Dometic makes AT LEAST two different 13,500 btu models. One is a 57915. 521/526 and the other is 57915. 621/626. For the purpose of this post I'll use the 37915. 621/626 because that's what I have on my 96 Aljo 5er.
Second, lets gather the information on RUNNING amps from the manufactures book.
Compressor running amps are 11. 0
Fan running amps are 2. 5
Total running amps are therefore 13. 5
Locked rotor amps are 54. 4
Now here comes the fun part. Airconditioners run on AC and not DC so you can't use ohm's laws to calculate the load because you are dealing with reactive inductance and capacitance and impedance instead of a resistive load. To make this story somewhat shorter - the unit is going to draw about 1400-1600 watts when running. This of course DOES NOT take into consideration the decrease in efficiency and increase in current when it get hotter outside OR when the unit first STARTS. Which takes us to our third point.
Starting current - that's our biggest problem here. As you can see a 2000 watt Honda unit will run a 13,500 btu unit just fine IF IT'S ALREADY RUNNING. However getting it started is the problem. Airconditions use capacitor start induction run type motors. I'll try and make this as simple as possible. Our electric AC motor has a locked rotor amp rating of 54. 4 amps. Now our rotor isn't locked (unless it's defective) but as far as the MOTOR is concerened during the first revolution or so the current required to get it spinning is very high. We are talking about milliseconds here and some phase shifting is going on and that helps decreases the current requirements BUT it still takes a lot of current to start an airconditioner.
So our big concern is not running current or watts but STARTING current or watts or SURGE CURRENT OR WATTS. Duo-therm by Dometic RECOMMENDS a MINIMUM generator size of 3. 5 kw (3,500 watts) for a single airconditioner in their instruction book.
So do you need a 3. 5 kw generator. Well maybe yes or maybe no it depends (don't you just hate double talk

Some generators have a very high SURGE watt rating and could start the aircondioner just fine. Others may not have a high SURGE rating and may not be able to get the unit started.
A Honda 2000 is a fine unit but is PROBABLY too small for a 13,500 btu AC unit. A Honda EU3000i would probably work fine BUT I would try it on a hot day at the dealer with some lights on inside the trailer before I bought one. Remember, if the vent fan is on in the bathroom, lights are on over the stove and the stove vent fan is running you have taken several amps (or watts) away from your available starting current.
My old motor home had a 4kw onan which would start and run the AC and microwave at the same time EXCEPT on a very hot day (110+ degrees). My son-in-law has a Tioga motor home with a 2. 8kw onan and it will ONLY run the AC unit and a few lights. Remember it's not so much the running current it's the starting current. There are some things you can do to limit starting current like turning on the fan portion of the unit first and then let it come up to speed before starting the compressor. Remember, fan first then compressor.
So, something around 3kw sounds about right if it's not too hot and your not trying to light up the whole forrest at the same time your cooking hot dogs in the microwave
Happy camping