"I don't really want to keep the entire rental paperwork for months and then have it go missing. My insurer seems to want those in original paperwork when I asked them about it."

So either:

1) Rent a safe-deposit box, and store the rental paperwork (and your other important papers) in it; or

2) Initiate a claim with your insurer now, and furnish them with the documentation you currently have. You can supplement the documentation down the road, if and when Enterprise sends you a demand letter.

I'd be truly surprised if your insurer insisted on getting your hard-copy originals, as opposed to a faxed copy of the original, or an e-mailed scan of the original. But I suppose that anything is possible.