I'm thinking coating the insertion tip with some RTV/gasket maker should fix things. Permatex black is some pretty strong stuff when cured.I’d just wrap it with electrical tape or something until it fit the way I want.
Having done a tire rotation in my garage, I would not jack without a puck. Things can shift a bit while jacking and having no puck would be very unforgiving. Even if a puck is loose, I'd still use it over nothing.