I ultimately devised another solution. It certainly isn’t perfect, but, in this case, it fulfilled my requirements and was a lot better than rewriting the whole Chatter component from scratch. I started by creating a new Visualforce page with the standard style sheets enabled and containing only the Chatter component. Then, in the original page, I included the new Chatter-specific page in an IFrame element. This effectively loads the Chatter component as an isolated element with its own style sheets.
In an ideal world, Salesforce would provide a standalone Chatter component that doesn’t require their standard style sheets to be included for the whole page. (If anyone at Salesforce is listening, please do this!) But, in the meantime, this is a reasonable workaround. I hope that by sharing my solution it will save someone else the time it spent for me to figure it out. Check out the sample code below, and if you have any questions, or if you have a better solution, be sure to leave a comment.