Probably too late to help you, but maybe others are curious too. I did Barr to the summit this morning.
Up to A-Frame, there's virtually nothing. The mile after that has some snow but it's no problem. The last two miles are more dry than not, but there are a few large snow fields. I followed previous footprints and got to the top without issue in trail running shoes. Definitely doable, but also definitely snowy and slightly tricky in a couple spots.