I figured out the issue. Apparently the settings I was using in a plugin on both websites was slightly different for each and was causing the issues. I still don’t understand how that was affecting the BP functions but everything works fine now.
I should add there are several other issues with links not working within BP pages. Such as: the links under posts (Comment, favorite, share), the links used to sort posts in the activity stream, and the read more link on posts in the activity stream (to name a few). Sometimes these links work and sometimes they don’t. The “post” link however, never shows up.
I have another site that is almost exactly the same with none of these issues. The only difference I can find is that the site having issues is an addon site on my server and so it is stored in a sub-folder. Could there be an issue with how BP pages load when they are stored in a sub-folder on a server?