After some searching online, I managed to find a blog post where somebody built an HV programmer to fix pretty much exactly the issue that I’ve had as well. While programming their AVR chips through a USBTiny, they accidentally set the reset disable fuse, locking themselves out of the AVR programming. They point out exactly the frustrating part of that: the microchip is perfectly fine – it absolutely works, and could be used still. It’s just that SPI programming is unavailable. Now, the post is from 2008, so the claim that there are few to no HV programmers are available might be quite outdated (I hope so, at least). I’ll first try to find a cheap programmer, but at least I now have an alternative available as well.
EDIT: In fact, I’ve found another blog post with an even easier solution. That looks like one I’ll put on a bit of perfboard, as I doubt this will be the last time I run into this problem.