We always called this thing a slide whistle and I own ones in plated metal (chrome) and plastic.

I don't know about other people, but when faced with the first problem on a different instrument, I used waxed dental tape (wider than floss).

Neither of my whistles need grease. If it doesn't slide easily I would think that wax would be safer than grease. I don't know if bakelite is at all absorbent, but if any of the components do take up moisture, the grease might penetrate and create problems down the road.

Again, I have never dealt with bakelite and bone, but warm soapy water and a rinse is usually good, and you might want to purchase a flute swab to clean the interior and to use on it after playing it.
Blessings,
Barbara