Proper method to attach horizontal board to beam posts in outdoor patio?
Your 2x10 idea is just fine. I wouldn't use hangers, though. They'll be ugly. 

I'd do this:


Cut a 2x10 beam to length, fitting it between the posts. 
Mount pairs of slotted bed rail brackets to the back side of the board, flush with the ends, one high and one low.
Set heavy screws as appropriate to engage the slotted brackets. 
Paint the board and hang it. 


Either that or simply mount it to the outer faces of the posts with nice screws. If you fit it well, that would look fine. It really depends on how deep you want the tv to be.