I am in the process of Buying a Ram 1500 diesel, and one of the things I looked at was the weird size tires that the Ram comes with. You are correct, the only LT tire in that size are the Nittos. That is why, I am planning on getting the Outdoorsman with the 17" LT tires standard. Then, I have much more to choose from. Just waiting for the $$$ if you know what I mean.
Difference between LT and P rated is that the LTs are more stout and heavier. A big complaint about towing with P rated is the Jello feel from the tires. LT tires get rid of that feeling.
Do you need LT tires? Probably not. But, you will probably feel a lot better with them.
Could you possibly get a set of P rated snow tires for your current rims and get another set of rims, maybe 17" or 18" with a similar to stock sized tire as a replacement?