Most common and correct is Chrome-Browser.
der Browser Chrome would also be possible. But it is rather formal and uncommon, would sound old-fashioned and maybe not even be considered correct.
Chrome Browser would widely be considered wrong. Traditionally, in German, compound nouns are formed by glueing the compartiments together, sometimes with a hyphen. ...
The correct version is:
Wir mussten leider feststellen, dass der Chrome-Browser einen Bug hat.
Here, we construct a compound from a browser further specified as Chrome.
Another possibility is:
Wir mussten leider feststellen, dass der Browser "Chrome" einen Bug hat.
Here, we're adding the name to the abstract term, and that's why I used quotes ...
What you're describing is a problem language learners always face, that it's not enough to just learn vocabulary and grammar. If you just translate each German word into an English word, and rearrange them to fit the rules of English grammar instead of German grammar, the result will not be a proper translation; it might not even make sense. The best you can ...