No need to actually do KS, just organize it as a group buy: If enough people commit and put down a deposit, the run is funded and it happens. If not just unwind it.
It’s a fundamentally different problem because you’re not, from either side really, worried about the classic Kickstarter “funds raised but we can’t keep our promises” issue. You probably already have a working relationship with a print house, print ready source files, and on the financial side you probably know exactly what all your cost basis is, e.g. setup costs of $x, $y printing costs per copy and fulfillment costs of $z. Kickstarter’s fees aren’t nothing.
It might actually be easiest to run such a second printing as a second kickstarter campaign.