the trail is relatively flat. it has few spots that may prove a bumpy ride for the trailer, but it shouldn't be too bad.
why not just walk? i personally suggest that you leave the bike and cooler behind and enjoy a nice trek out there and do your day hikes from there. it is not too far from the trailhead at all.
we don't need no steenking coolers!!!
you going to bike out a lounge chair and a tv too?!!
just kidding! just trying to talk you into backpacking the right way. hcsna is an awesome place, you'll love it out there. hope you have a great time!