Yeah, so I think this is pretty much the a kind of scenario the feature is
designed to break - an essentially stock browser navigating to some site, that can then wrap a trusted site, and potentially be a stepping stone for phishing or other nastiness. Obviously, if you have detailed control over the client - everything goes! But for fairly standard browsers, you don't want to allow this kind of nesting; at least not when the two sites aren't cooperating (and clearly linkedin isn't).
It's possible there's some addon api to work around this, otherwise you'll need to use a proxy - which isn't too hard, really. Essentially: you need to have admin level control over the browser to allow this, or some other proxy; the browser won't allow a random website to see another site's cookies, even indirectly - which is by design.
So in your case: since this is a feature, not a bug - a trivial workaround looks unlikely.