I want to thank you, @Tanaike, for your response (I don't have enough 'points' to upvote or comment, so my only option is an 'Answer')
I know this thread is several years old, but I thought others might be interested in my personal experience.
First of all: "Workaround 1" worked for me!
The function/method "Sheets.Spreadsheets.Values.get(spreadsheetID, RangeName).values" was giving me an "missing a valid API key" error, so I swapped it for "sheet.getRange(RangeName).getValues()".
Most of the above was set as Global Variables, i.e. outside of any functions.
Weird thing was that this error occured only when running from within the [container] sheet, not from the "embedded" script.
For instance: If I had an active onOpen() function, and I opened/refreshed the Sheet, the script would log a "The request is missing a valid API key." error, and the UI/Menu I had built therein would not load.
I could, however run the onOpen() function from within the script itself, and the menu would appear, and function, within the Sheet. If I disabled/renamed the onOpen() function, and reloaded the Sheet, I would not get the error message.
Makes sense, as the simple loading of the Sheet does not appear to run the script, but when one does access it (the script), i.e. through the onOpen() function, then the initial global variables are read (and the error occurs).
However, when I ran the same function, or others, from within the script itself, they would run ok. This permissions conundrum is what has led me on a wild goose chase all over the Internet, ultimately landing here.
All this after numerous other issues, in the course of whose resolution I built a Google Cloud Project, added APIs (e.g. for Sheets), added scopes to the oauthScopes section of the manifest, and more.
It was only after I made the replacements described above that everything worked, both from the script, and its container spreadsheet! So, THANKS!
Anyway... Sorry for the long post, but I hope others may benefit from your solution, and in which context it helped me.
Cheers,
-Paul