How to write proper nouns such as company names whose spelling is nonstandard in reports?
I challenge your contention that these things are spelled incorrectly.  When you write 'Toys "R" Us' you are correctly spelling a proper noun.  Names are signifiers, and the entity who controls the name controls how it is correctly spelled.  I would not have told my high school friend whose last name was "Tomson" that his name was spelled wrong just because for most people it was spelled "Thompson."  Likewise, 'Toys "R" Us' has chosen a particular spelling for its title, and that's the correct name.

So now comes the question of how to communicate such titles, and here, I see four basic cases:


Just use it as written, and count on people to understand: not great with Toys"R"Us since quotes are semantically loaded, but fine for a well-understood "misspelling" like Google.
Often, there are commonly used variants that are simpler.  If you say Toys R Us, people will know what you are talking about even without the quotes, so its OK.  Same with the artist formerly known as Prince. If you say Toys Are Us, however, you've gone a step too far and "corrected" the name into something incorrect.
Put it in quotes, like I did in the first paragraph: 'Toys"R"Us' (here I made the unusual choice of single quotes because of the presence of double quotes in the name).
If you think it is still strange enough that you think people will think you misspelled it, you can add [sic] afterwards, as in "Tomson [sic]"


I don't like italics, as a solution, personally, since I generally see italics used for emphasis or for definition, and neither is the case here.