Not advocating the use of automotive jumper cables for this purpose, but just want to correct a couple of points that I believe are common misconceptions:
They may only be rated for 12 - 24 VDC but perhaps they'd be up to this sort of challenge.
Voltage ratings of wire and cable are not for the conductor, but for the insulation around it. Current ratings are for the conductors, not the insulation. Since grounding conductors are often provided uninsulated, I would think that the voltage rating of automotive jumper cables should be irrelevant in a grounding application. You just don't see voltage ratings on bare wire.
CCA is generally only good for high frequency or low power stuff. DC and AC power just aren't the same.
And copper-clad aluminum has been used for 120/240/etc.V//60/50Hz wiring in the past, and may still be permissible under NEC in certain applications with listed connectors. Why would someone use CCA? 1) because aluminum is less expensive than copper, and 2) because corroded copper has better electrical properties than corroded aluminum. Uncorroded aluminum is a decent enough conductor; the bad press that aluminum wiring has received in the past is due to its properties when it corrodes and also its softness that contributes to loose connections.
As far as I know**, the NEC hasn't approved any spring-clamp devices for temporary grounding. And if they did, I'd suspect that a welding ground clamp might be a better choice than an automotive jumper cable. I believe any inspector would approve stage grounding using the same fixed "pipe grounding" clamps required for permanent installs. Installing and removing those clamps really doesn't take a lot of time; just a little more than installing and removing a lighting fixture clamp from an overhead truss. And getting a spool of #4 AWG copper wire isn't THAT difficult or expensive in the grand scheme of things.
If you are renting a stage, do you really think the stage provider is going to appreciate you scratching through the paint to get a good ground? Better talk the issue over with them first.
**I haven't studied a recent copy of the NEC.