From French history there is the sans-culottes movement denoting the anti-aristocrat, anti-bourgeoisie working class who wore regular long pants. If you wanted to invent a term that meant anti-sans-culottes, i.e., an aristocratic-bourgeoisie counter movement term, could you say avec-culottes or maybe des culottes perhaps?
No, you can't use those names; the people they opposed (violently) were simply the royalists (réf.) or more generally the aristocracy but if we were to speak of an opposite it would be figuratively not literally. Moreover they were not the only opposition to the royalists: they were rather the grassroot (réf.), the Girondins and the Montagnards having variants ideologies that sprung from the sans-culotte's credo and they too made a strong opposition to the royalists. In the French language you do not consider those terms as real opposites. The term "sans-culottes" stems from the fact that the royalists did wear "culottes", and that those garments were rejected by the revolutionary section of the population that came to be known as "sans-culottes"; they chose instead to wear trousers (pantalons).
I can't think of any expression based on culotte that would work (beyond anti-sans-culottes that you already used).
Both les culottés and ceux qui portent la culotte already have specific different meanings.
Here is a couple of ideas:
Last year, some opponents of the gilets jaunes (not really the same spirit than the sans-culottes, but sharing some anti-establishment views) called themselves the foulards rouges (red scarves). We might extrapolate from this and use les foulards blancs (white being the color of the royalty) or les foulards de soie (silk scarves). One issue is the foulard is more a female garment.
Another suggestion could be based on another aristocracy/bourgeoisie clothing that disappeared (along with the heads behind them...) during the French revolution and call the anti-sans-culottes: les perruques poudrées (the powdered wigs)...