The jib crane is extremely light duty. It looks to me like the I beam has a twist in it (?) further significantly de-rating it. All of the moment loads go thru the base, and then have to be transferred to the structure, its rotating so the moment is applied in different Z-rotations. Some data...I have a 1/4t jib crane in my shop, it has a 4x4x4 block of concrete in the earth which supports the overturning moment and bears against the 2000psf soil on a 16sqft face on each side.
Much better solution imo, a twin post gantry which rolls over the opening and transfers loads vertically down to the structure. If 3 decent sized guys can stand on each side of the opening you are already there structure-wise. The place where things could get squirrelly is if the gantry casters decided to roll into the large opening leading to a very bad result. How I would approach that is with non-swivel casters with a V-groove. Then you screw down sections of angle iron with the corner "up" as a set of parallel tracks. Make sections of the track removable (screws, etc) so that you don't have to trip over them forevermore.
Single phase electric chain hoists are out there if you poke around enough. Make sure you have enough chain in the bucket to reach the floor.
I wouldn't limit myself to lifting loaded pallets, I would rather bolt metal tabs (lets say 1/4" x 2 flat bar with holes punched in the ends or eyebolts) to a part, use soft-slings in a choker config, and simply load an empty pallet on the 2nd floor.
If you were bound to pallets nonetheless, then I would "invent" aka custom-design and fabricate an end effector similar to the way a boom-truck picks up a pallet of shingles to sky it to the top of a roof peak for roofers to unstack. Essentially its a large, balanced C-frame with forks that slip into the pallet and have a center-of-mass pickup point overhead. You can match this to the overall headroom available to know how high you can load the pallet.
If I had a barn that nice I think I would build some fantastic bathrooms nice as a Bucc-ees and host weddings or parties lol. Not fill it with tools and other ferrous hoardings.