Yes, but it's Dumb.
Bardic music specifies by RAW that the music must be audible. Attach bells to your dancer, and by RAW, you are doing it. * cha cha *
That's easily dealt with.
More problematic is that Bardic Music'ing is a standard action, and while it does linger for 5 rounds after you do it, that's lame.
There is no reason why your chaos monk/bard should not be able to inspire courage as a free action so long as his other actions are spent moving and hitting things with capoeira. That's so fine I have no words. Houserule that right this instant.
But, if it must be RAW and only RAW:
Check out Song of the White Raven from Tome of Battle: Book of Nine Swords. With a 1 level Warblade dip (which gives all kinds of goodies to a melee combatant), you can take this feat that allows you to start a Bardic Music as a Swift Action.
Then, check out Lingering Song, the feat, that makes Inspire Courage last ten rounds after you start it.
So at the start of combat you start a dance routine, and for the next ten rounds the party gets the effects of your Inspire Courage. Re-start the music as necesssary, and take Extra Music as necessary to keep that Courage flowing.
Note: You can do this with the Martial Maneuver, Martial Stance, and Song of the White Raven feats instead of a Warblade dip + 1 feat, but that's 3 feats, our of the 7 general feats you get your entire adventuring career. Seems like a bit of a waste just to do your one basic thing.
But that's overpowered! Inspire courage is super duper!
Inspire Courage is +1 to hit and damage. That is the same as a 0 level spell. If this guy is mixing monk and bard, even with the Ascetic Mage feat (which he should take), that bonus is unlikely to ever get to the +2 to hit and damage stage.
Compare this to a Barbarian's Rage, that gives +2 to hit and +3 to damage from level 1. Compare this to Elation, a 2nd level spell that gives +2 str and +2 dex as a morale bonus. Compare this to Enlarge Person, Glitterdust, or Fist of Stone, much less Bite of the Werebear or Greater Blink.
Given the fact that this is a t3/t5 multiclass, there is no reason - absolutely no reason - to ever think that Inspire Courage + punching people is overpowered. It is not. It is, if anything, underpowered. Compared even to a straight Fighter or Barbarian, neither of which is seen as very high tier except at extremely low levels.
Check out: Snowflake Wardance, and if your pal is lagging behind in terms of being able to do stuff, check out this thread.
Multiclassing Penalty - This is one of the least-used rules in the game, because penalizing multiclassing (free multiclassing is the absolute greatest strength of the 3.5e system) is dumb and leads to boring characters.