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My thoughts are that the [Printer...] button is intended for 'last minute' amendments to the printer driver as the image is on its way to be printed. (and as such- is temporary)
The [Page Setup] button (two in Macs) bottom-left of screen, is a better choice for setting the printer properties before designing the photo layout and paper to be printed.
If you set up the printer properties from [Page Setup] then create a User template (click + on the 'Template Browser' panel), you can always return to that setup from the Browser panel for future prints.
Print templates include the Printer properties setup options. (a reason why many beginners have problems- eg. selecting a LR default template for (1)5x7 which is designed for 'Letter' paper size)
I would assume that most printer driver dialogs also have an 'internal' option to 'save' a properties setup.
More experienced 'Printer people' may have a better answer than mine.
Stupid me, I didnt even notice the "Print Settings..." button in the lower left (I'm a recent convert from Aperture)
That dialog has a "Save" instead of "Print" which, of course, explains everything! is the answer.