In the following sentence, how would I indicate possession if the word "business'" were replaced by the name of the business: like "Fry's" or "Wendy's"?
Some business' employees are happy.
It seems strange to say that Wendy's employees are happy, since I'm referring to the business, and not just Wendy.
I would just say some Wendy's employees are happy. that construction does not take the possessive; compare some Stackexchange employees are happy (not * some Stackexchange's employees are happy).
The noun would have to be possessive if you were to add of, as in some of Stackexchange's employees are happy. The solution, of course, is not to add of.
In other contexts where the possessive is actually needed, I would probably just use the company's name as is, even though it isstrictly speaking illogical, or rephrase, if it is not too cumbersome:
Wendy's most popular menu item is ...
The most popular item on the menu at Wendy's is ...
I suggest simply:
Some employees of Wendy's are happy.
to avoid the problem altogether. Otherwise, I think it would have to be "Some of Wendy's employees are happy" (to be possessive) and that does give the impression that they are employees of Wendy, not of Wendy's the company, as you say.
That said, the possessive of McDonald's is frequently referred to as McDonald's https://www.google.com/search?q=%22of+McDonald%27s+customers%22 (but mostly by journalists and not by McDonald's themselves).
...22 percent of McDonald's customers said...
Perhaps that is simply because McDonald's is so famous that nobody is concerned about any misunderstanding that that means "the customers of McDonald". However, originally, it would have meant "the customers of [the] McDonald [family]" so it is not completely misleading if taken to mean that anyway.
I think the rule would be that, as McDonald's is already a possessive, no further apostrophes are needed to talk about its possessions. It can be read as meaning something slightly different to what is intended now that the possessive is a proper noun in its own right but McDonald's' would be regarded as wrong so it's the only option there is (or to avoid it by using McDonald's as adjective rather than possessive).
You could add the registered trademark symbol to it so it doesn't look like the employees belong to someone named Wendy rather than the actually franchise
Regardless, Wendy's is already possessive, but it isn't used so. It's used as an adjective to describe "employees".
Good employees
[Business name here] employees
