1) Found an old USB disk, fittingly sporting a Batman logo.
2) Formatted it as a FAT (MS-DOS) drive.
3) Downloaded the free ISO of Darik's "boot and nuke" data wiping tool (DBAN) from http://www.dban.org/
4) Dowloaded Unetbootin from https://unetbootin.github.io/ and launched it.
5) Used this to turn the Batman disk into a bootable disk based on the DBAN ISO.
6) Due to a bug somewhere in this pipeline (Possibly Unetbootin? Or DBAN itself?), I needed to edit the "syslinux.cfg" file in the root folder of the USB disk. This entailed replacing all mentions of "ubninit" with "ISOLINUX.BIN" (and one mention of "ubnkern" with "DBAN.BZI"). Attention, case-sensitivity going on here. (This handy guide helped me to figure this out. It also points out that it is a good idea to remove the "--autonuke" flag in this config file since that leads to a potentially very dangerous auto-nuking USB stick...)
7) Set the BIOS of the target laptops to prioritise booting from USB drives.
8) Plugged Batman into the laptops and them started up - choosing the aptly named "Default" as wipe option.