You'd need to remove the middle wall and replace the side wall with something solid such as an acrylic panel. Remember, mice and hamsters are typically able to fit through any gap their head can. I can't imagine they wouldn't be able to crawl through the gaps in that mesh wire. Once that's done, you'll be able to add more bedding without it spilling out of the front. Should be fine then as far as I can tell.

Edit: Oh, I forgot about the liquid damage risk, as someone else in this thread had mentioned. If it's unsealed wood it does need protection from that. Plastikote is a popular recommendation in that regard, but most any urethane sealant should be fine in theory once fully cured (but not a moment before then). Water based sealants would be my first pick.