While this is not "scientific data", I did find some empirical data on the subject from this website. The author talks about his experience with the Honda Pilot and its VTM-4 system:
Reading about VTM-4, the way I understood it to work was this...
Under normal conditions, the VTM-4 system is FWD biased and transfers power/torque to the rear as needed. If you need extra traction then engaging VTM-4 Lock would lock the rear differential and send power to both rear wheels.
What I experienced today was quite different. I found a nice little hill with some sections that would get a vehicle crossed up (For anyone that doesn't know, getting crossed up basically means a pair of tires at opposite diagonal corners are compressed and firmly on the ground while the other pair of tires are drooped and possibly not even touching the ground. An open 4WD/AWD system would simply end up spinning the two wheels with the least traction and you'd go nowhere.
So, what happened? I started out with the system on full auto and started up, didn't make it too far. Hit VTM-4 Lock and only made it a couple more feet. I had my wife watch what the wheels were doing...
Full Auto/No Lock - Only the front tire would spin, nothing going on in the rear.
VTM-4 Lock - The front tire at full droop and the rear tire at full droop would spin. The rear tire firmly planted on the ground did nothing and there was no forward progress. Turned off VTM-4 Lock, back to only the front wheel spinning.
Doing some reading, it sounds like the rear "lock" isn't a true mechanical lock but more like an aggressive limited-slip with clutches. That being the case, this situation and the results sort of make sense and I was likely just overpowering the "locked" clutches in the rear, however I was surprised that the rear tire wasn't spinning at all when in full-auto. What it ultimately felt like was that there's a center differential with clutches, splitting power between the front and rear but allowing slippage. Engaging VTM-4 Lock would lock the center diff to send equal power to both front and rear differentials but leaving the rear differential unlocked, performing like a part-time 4WD system with open differentials front and rear. If VTM-4 not locked was sending power to the rear I wasn't seeing it.
With this in mind, it appears your VTM-4 system is working as others have seen it work. I think you are over stressing the system with your hill climb and extra load.
I would highly suggest you get a vehicle with a higher towing capacity, like a truck or large SUV which can handle this better. If the information I'm seeing is correct, the towing capacity is 4,400 lbs. When you look at this number, it's not only the weight of the trailer, but the weight of all equipment, fuel, and people you are carrying as well. The trailer may weigh 2,200 lbs, but with the addition of two grown adults (300-450lbs total - throw in kids it's even more), equipment (500-700lbs), fuel (~100lbs), etc, etc, etc. This adds up quick where you can quickly max out your vehicles capacity. Now, throw in that you are going up a hill (steep, as you said) and you have a further burden on your vehicle.
I'm not going to tell you which vehicle to get, as my opinions would quickly get in the road. What I will tell you is to get away from an AWD vehicle and get a 4WD vehicle. You might now ask, "What's the difference?" ... The difference is, the AWD system is full time on, while the 4WD system is selectable. The big thing you are looking for with a 4WD system is the availability of a low range. By utilizing a low range, you effectively double (depending on the system, this will vary ... I'm using "double" generically) the amount of torque your engine is putting to the ground, leveraging mechanical advantage to do so.
The second thing you need to make sure the vehicle has is a true locking differential. It does not sound like your Honda has one. If you are curious, do a test as described in the excerpt I provided. It is a very good suggestion without putting undue stress on the vehicle your are driving. Ford, GM, Dodge, and Toyota (to name a few) all utilize this type of differential in their trucks. It will be mentioned on their window sticker if new, or if used, you can easily have the VIN run to see what options were installed on the vehicle when new (take the VIN to the corresponding manufacturer's parts department ... they can pull this information for you - if they are willing. I said "can" ... doesn't meant they will, lol.)
The third thing I'd look for is to ensure the vehicle is a true rear-wheel drive. Most large trucks/SUV's are setup this way. Rear-wheel drive is going to be a lot more effective while towing a vehicle than front wheel drive. This is because the weight of the trailer is providing weight over the main drive axle, which means you'll have better traction.
The other thing which I would tell you to do if not doing so already, is your driving technique. Ensure you are not "balls-to-the-walls" when going up the hill. You are again, just causing undue stress on your vehicle. The slow approach will provide you more comfort on the way up, as well as make it easier on your vehicle. You are trying to avoid wheel slippage if at all possible, which I believe may be at the root of the issue to begin with.