What is the central conflict of the play Travesties?

The central conflict of the play Travesties rests on the antithetical virtues of Dadaism and Aestheticism. This main conflict is played out against the backdrop of the divergent natures of capitalism and socialism.
In the play, Carr argues for the rationale of Aestheticism, while Tzara proclaims the virtues of Dadaism. Like so many Dadaists, Tzara supports art as a means of pushing back against the dictates of the capitalist bourgeoisie class. In Dadaism, the central argument is that art belongs to the common classes as well as the ruling elites. Essentially, the Dadaist movement was a rebellion against the tenets of capitalist art.
Dadaists argued that classical art in capitalist society was restricted to the few talented individuals who could produce it. These individuals were invariably feted by the upper classes, and their works were touted as the epitome of artistic excellence.
Meanwhile, the working classes had little access to this exclusive sphere because commercialized art was entirely dependent upon the classical rules of aestheticism. The reality is that few individuals in a capitalist society could lay claim to having mastered these rules. However, the common artist was less likely to acquire the material benefits associated with patronage by the upper classes. Hence, the Dadaist movement was predicated on the discontent of the working classes.
Dadaists argued that modern civilization suppressed creative self-expression. They proclaimed the right of everyone to undertake creative endeavors. Certainly, Dadaist art was not necessarily good art (as defined by the classical rules), but they allowed the common man to give voice to his reservations about the power structures of his time. You can see an example of Dadaist philosophy in the beginning of the play through Tzara's nonsensical limerick. 
Essentially, the nonsensical or irrational elements in Dadaist art was an anti-capitalist expression of discontent. In Tzara's time, many disaffected artists (most of them draft dodgers) sequestered themselves in Zurich. They argued (like Stoppard's Tzara in the play) that the Great War was predicated on greed and cunning rhetoric, designed to keep the masses in ignorance.
In the play, you will see the Dadaist (socialist) conception of art pitted against the capitalist notion of aestheticism. This is the central conflict of the play. Here is an example:

CARR: The nerve of it. Wars are fought to make the world safe for artists . . . the easiest way of knowing whether good has triumphed over evil is to examine the freedom of the artist. The ingratitude of artists, indeed their hostility, not to mention the loss of nerve and failure of talent which accounts for 'modern art,' merely demonstrate the freedom of the artist to be ungrateful, hostile, self-centered and talentless, for which freedom I went to war.
TZARA: Wars are fought for oil wells and coaling stations; for control of the Dardanelles or the Suez Canal; for colonial pickings to buy cheap in and conquered markets to sell dear in. War is capitalism with the gloves off and many who go to war know it but they go to war because they don't want to be a hero. It takes courage to sit down and be counted. But how much better to live bravely in Switzerland than to die cravenly in France, quite apart from what it does to one's trousers.
https://brooklynrail.org/2006/7/art/dada2