If you are going to deflate your tires to move the trailer under something low, make sure to reinflate them right away. Don't store the trailer on severely underinflated tires.
Why not just raise the shelter? Maybe use a pry-bar to raise one end while you move the trailer, then lower it again?
Have you seen where I store my 17'? Linky.
If you are strong enough, and you are storing on a reasonably level, reasonably hard surface, you can move the trailer by hand with the hitch wheel. If you crank the hitch jack as low as it will go, you get some extra overhead clearance.
If moving by hand, you might want to think about how you can engage the emergency brake. The idea just occurred to me, so I have not thought it through.