Supplementary Internal Filter

You would need to run both filters for at least six weeks. But remember that by the end of that time the bateria would be split between the two filters in proportion to their size. If the external had, say, five times more media than the internal that would give you a ratio of 5:1 external:internal, so the internal would only be able to support a sixth of the fish currently in the tank. And if your intention is to put the internal in another tank, when you moved it you'd only have five sixths of the bacteria left in the external.

If you wanted to cycle it a bit quicker than that, in around a month, you could swap some new internal filter media with mature external filter media as having mature media actually inside the internal would seed it better. Put the mature media first in the direction of water flow so that any dislodged bacteria would get swept into the new media.

I wasn't sure what your intentions with the internal were, it didn't sound like you were intending to cycle it for another tank, but I though I'd better mention it just in case.

Yes, since it is safe to remove up to one third of the filter media, if you took the internal out the bacteria would make up the difference so quickly you'd probably never see any ammonia or nitrite - assuming your internal has less than around a half/third of the media the external has.

If you are wanting the internal for extra circulation you could always run it empty. But if you want to fill it I'd go for filter wool rather than cotton wool - I use that round the home and I would worry it would try to soak up tank water rather than allow it to pass through and you'd end up with a horrid gooey mess. I'd try plain ordinary generic filter wool, it's fairly cheap. That would grow bacteria on it, but depending on what your tank is like you might find it clogged fairly quickly so need replacing before it built up bacteria in any significant numbers.