The following won't help you with your current project because, as MikeWalsh pointed out, Fossapup does not have a new enough version of glibc*.
As bigpup noted, Puppys do not use apt. For your future projects, you might want to consider installing pkg-cli, viewtopic.php?p=34701#p34701 if not already present. I think it's already built into Fossapup64. The above post provides a link to its 11 page discussion on the 'old forum'.
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* glibc libraries are 'foundations' --part of the framework upon which other applications depend. Upgrading glibc may result in 'breaking' many of the applications which depend on the earlier version.
This thread, viewtopic.php?p=4251#p4251 may still lead you to a downloadable version of palemoon tweaked to incorporate within its structure, and isolated from the OS's built-in glibc libraries , a more recent glibc version than those builtin. It was a project involving the efforts of Walter dnes, watchdog and Mike Walsh, based on the pioneering work of battleshooter. viewtopic.php?p=13146#p13146
You can download a version of the tweaked-palemoon, dissemble it and see how that isolation and usage was accomplished.
IIRC, in replay to my assertion that 'an OS can only have one glibc version', MikeWalsh suggested that that might not be the case. I concede he may be correct. But with two 'builtin' versions, some routine would also have to be established so that the proper glibc version would be used with each application. Off hand, that seems more complicated than running one version of Puppy from another by employing qemu or virtualbox.