Trademarks are registered within a specific category of business. That's why Apple, the music label, could exist next to Apple, the computer company, all those years back. It's also why Apple got into a trademark dispute all those years down the line when they got started in the music business.

The http://argos.com/ people don't have trouble with the ARGOS trademark (there was a lawsuit about this!). OP's product might not be so lucky, though.

If OP can get a trademark registration for the right category through, they should be fine using the name Argos from what I can tell. Either Argos or ARGOS might object to that trademark and make that very difficult, though.

That’s not quite how it works. Companies argue that there’s no brand confusion in order to coexist with similar trading names and use market sector as evidence but there isn’t any specific law that says this has to be the case.

To use your Beetles vs Apple Computers case study, Apple Computers actually settled with the record label and part of the agreement of that settlement was that Apple wouldn’t enter the music industry. It wasn’t a case that the record label lost, it was a settlement. Hence the follow up case between the two after Apple released iTunes (or whatever it was that triggered their “arrival” in the music industry).