I downloaded the lastest 'solaar' SBo package, installed it, when I run solaar I get: Does this mean python is looking for a moduled named solaar.gtk?

I searched the Internet, found no useful responses.

solaar.gtk is most probably a submodule of solaar.

It isn't part of the Solaar package. I haven't found it elsewhere. One of the hits of a search says it isn't, but offers no alternative.

https://archlinux.org/packages/extra/any/solaar/
/usr/lib/python3.11/site-packages/solaar/gtk.py is a file in the package. But I don't know what did you download and if that gtk.py is really there. Or you have made a mistake somewhere.
You can find it here too: https://github.com/pwr-Solaar/Solaar.../solaar/gtk.py

On one of my installations I ran into OP's situation. I had chosen to install solaar without solaar-gtk that also popped up in my search and the gtk pkg was not listed as a dependency of solaar. Once I installed solaar-gtk I had no problem with solaar. But solaar-gtk lists solaar as a dependency so if you go to install that first thing it will bring in solaar automatically.

Not sure which of my 4 installs it is but I remember the description of the gtk version being so vague I thought it best to try just solaar but that particular DE required the gtk pkg to function.

I downloaded the solaar package from slackbuilds.org. I have gtk.py the same place you do. Is it not being found? Is there a defect in my environment?

Hmmm... The only hit I got when I searched on "solaar-gtk" is this message. There's nothing in Slackware or on slackbuilds.

Solaar installed its libraries in /lib/python3.11/site-packages; all my other python software installed them in /usr/lib64/python3.11/site-packages. After I moved them solaar runs. It still doesn't find my Logitech unifying transceivers and their attached devices - but that's a different problem.