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I am a regular user of LaTeX (mostly through Pandoc). There are different methods for installing TeXLive:
Through the Debian repo with apt
By downloading the cli installer and installing over the web
Downloading tarballs
etc.
While I have installed it through apt, and it works without a hitch, is there a more Puppy way of doing this that I should try?
NB I've only been using Puppy for a little over a week (but I'm a longtime Linux user).
To Linux software! 
It really depends on the Puppy version and what it uses for installing added software.
If the Puppy version has apt package manager.
Using apt is most likely the best thing to use.
The Puppy Package Manager(PPM) in a Puppy version, will provide software for that specific Puppy version.
Most all Puppy versions can use deb packages to install software.
However, the deb package was not specifically compiled for Puppy Linux.
so it may or may not work without having to do other things to get it to work.
All Puppy versions can use pet packages to install software.
This is a topic you need to read about adding software to Puppy Linux:
viewtopic.php?t=1819
The things you do not tell us, are usually the clue to fixing the problem.
When I was a kid, I wanted to be older.
This is not what I expected 