Try using CssRewriteUrlTransform when bundling:
bundles.Add(new StyleBundle("~/Content/css").Include(
"~/Content/bootstrap.css",
"~/Content/body.css",
"~/Content/site.css",
"~/Content/form.css"
).Include("https://stackoverflow.com/questions/22700385/~/Content/font-awesome-4.0.3/css/font-awesome.css", new CssRewriteUrlTransform());
This changes any urls for assets from within the css file to absolute urls so the bundling doesn’t mess up the relative path.
Docs for CssRewriteUrlTransform