What’s best way to change the Style of All Viewports across all Pages and through grouped Viewports in a Layout Document?
Change the style for all of the involved scenes in the SketchUp model and update the reference in LO.
Unfortunately, it will not work when in LO a different (changeover) style was chosen than in the scene with SU
Of course it won’t. I didn’t say it would. Resetting the style for the viewports will be needed.
You also didn’t say it would not work either in this case 
Thinking of the skills I know the OP already has using LayOut there seemed to be no point in treating him as if he didn’t already know that.
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Having knowledge of OP knowledge and yours, you should write that the best way to change the style in all viewports on the sheet is to press Ctrl + A, then go to the MODEL SKETCHUP tab and change the style in the Style Tab.
Before:
After:
Since your advice requires resetting the styles, I think the above solution is better. They should be applied to each sheet.
Thanks for the Help. We have been selecting all on the page but there a lot of pages and Grouped Stacked Viewports so we were looking for a new way. Both worked, and were Similar in Effort. Changing Style in the Model was a bit Quicker but not by Much. (By the time you Save a Separate File, Change Style, Relink, Relink Back to Original, File Management, Waiting on my Machine, Etc) In the end Both Work. Thank you very much. This is all part of our DWG Export Process which I should Post for Reference in the DWG Export with layers Topic.