I'm running an up to date Manjaro system, and although I have kernel 6.1.9-1 available as an option, it's not installed and it's not listed as LTS nor is it "recommended".

Does this mean that'll change soon with an update, and I should start running this kernel?

You may want to move away from Manjaro in general[0], but I would advise against doing anything that the OS doesn't expect you to do - so I'd wait for an update.

[0] https://github.com/arindas/manjarno . EndeavorOS seems like a pretty good "Arch but easy" distro these days, and maybe Manjaro's improved since the last time I ran it (a few years ago now)