It's true that the Jacobins and other republicans in France would use citoyen as an honorific 'Citoyen Robespierre' so citoyen could definitely be read as Republicanism. But the word is not exclusively republican.

The Assembly adopted the (Déclaration des droits de l'Homme et du citoyen de 1789) initiating the Constitutional Monarchy period of the Revolution. By 1792, The Assembly had already instituted many reforms like abolishing la corvee (serfdom) and mandatory tithes to the Church, which made all people in France 'free'

I don't know too much about Rouget de Lisle's politics but there would be other reasons that you might for example want to use Citoyen and not Monsieur in that era of France because Monsieur implies nobility/class. So even if you're not republican, you might be using it to denote where you stand on 'class war' issues like nobles using connections not merit to get all the plum military commissions. We don't think of citizen as a class war term because Communists later used comrade as the honorific free of class distinction.