How to repair wood rot on exterior (decorative) exposed beam?
but what is the best way to repair this beam?


It depends how you personally define best. The absolute best (money and time no object) is to replace it.


  I'm thinking about removing the rotten wood


Before you start, be prepared that the rot may go a lot further than you expect (though maybe it won't of course).


  What is the best filler to use?


For small areas, Wood filler :-)

For slightly larger areas, Wood. Appropriately treated and/or painted.

If there is a lot of rot, I'd be thinking of remedial action more significant than filling.


  Or should I try something else?


For small areas, you might be able to use a wood hardener.

If it is non structural, you can scarf in a replacement section.

If it is structural, i.e. holding up your roof. I would consult a specialist, in my part of the world that might be a surveyor, elsewhere, a structural engineer maybe.