If you don't mind staining your camera with adhesive, you an adhere a small piece of velcro to the cap and to the bottom of the camera (for example) so you can stick it while recording.
Another option that will save your camera is to get a bolt that fits the tripod mount thread and use it to attach a thin piece of plastic, metal or wood and stick the velcro to that thing insteat of your camera's body.
If you bolt a steel or iron piece to the tripod mount, you can use a magnet to adhere the cap to it, most cameras won't mind a small magnet under them.
Yet another option is to get a small wrist band and use it exclusively for the lens cap. I'm thinking if the same kind that you would use with the camera itself, but my suggestion is to use a separate one for the cap, so you can hang it on your hand further from the camera. When you dont need it to be separated, just tie or interlace the two wrist bands together or join them ising a chain ring, a small clip, hook or something alike.