Actually, not paying those who conduct religious service is not the same thing as no state money whatsoever going to religious organizations.
Since 1959, the French government pays the salaries of teachers in private schools, most of which are religious, and gives subsidies directly to those schools.
So unless something has changed in that respect, the state does have leverage in that area.
Churches, temples and synagogues built in France before 1905 are the property of the state. National and municipal governments maintain these buildings, which are used free-of-charge by the clergy.
Surely getting all-expenses-paid buildings free of rent is nice too. There's probably not many of those buildings that are Muslim though.
Other mechanisms of leverage:
The law on “dangerous cults” simply grants the government the right, under judicial review, to dissolve such associations if they violate French law. The French government also retains the right to review decisions granting a special tax status to religious organizations under the 1905 Law of Separation if worship is not their “exclusive activity.”
And to be more precise Marcon even intended to amend the 1905 law:
Macron began a consultative process toward this end, stressing the need to set up an interlocutor for French Muslims (similar to those of other religious groups), create a framework for financing places of worship and collecting donations, and a system to vet and train imams working in France. Macron’s initiative sought to amend of the Law of 1905 on the Separation of the Church and the State (Law of 1905) with the goals of intrusively reforming religious organizations and ending foreign funding pouring into Muslim communities, which Macron felt prevented “French Islam from entering into modernity.”
[Macron] went a problematic step further, warning that he would make no concessions to a “political Islam…which wants to secede from our Republic” and appearing to say that he would cut off foreign funding for Islam in France.
Historically, France had accepted foreign-paid imams for the better part of the 20th century as well as other forms of forging funding:
Algeria, and Turkey considered themselves to be legitimate interlocutors with the French state on issues of religion. They sent paid imams to France, funded prayer rooms, and organized national federations to serve as conduits for their influence. [...] France understood, perhaps a little too late, that delegating the management of Islam to foreign powers was no longer sufficient and perhaps even creating new problems. [...]
During the second phase (1990-2000), the state exerted its control over French Islam through a representative council. [...] But that was not an easy task: Algeria, for example, was reluctant to give up its role as a caretaker of the Great Mosque of Paris. Then Interior Ministers Pierre Joxe and Charles Pasqua tried to build an Islam of France based on the supposed influence of the Great Mosque of Paris throughout France. For example, in 1994 Pasqua granted the Great Mosque of Paris monopoly authority over the certification of halal meat.
So there you have a concrete example how the French state tried to exercise influence despite the separation.
However, the creation of a French representative body of Islam has not ended the influence of those countries, partly because of the ineffectiveness of the CFCM, and with Algeria and Morocco fighting a proxy war for the allegiance of French Muslims. [...]
One of the few goals achieved in these efforts concerns the organization of chaplaincies. While Muslim chaplains are relatively well-organized in the army, they remain too weak in prisons and in hospitals and practically nonexistent in public education. Furthermore, foreign imams remain in France in high numbers. According to the ministry of interior, 151 imams have been sent by Turkey (which has undertaken a spate of religious outreach to Muslims across Europe over the past decade), 120 by Algeria, and 30 by Morocco. The official dialogue with these countries has made it possible to establish a rule requiring imams sent to France to follow civic and administrative training provided in French. The aim is to remind imams and chaplains of the rules they must follow, and to familiarize them with the history of laïcité, or France’s distinct conception of secularism.
For example, passing a law declaring some organizations or even persons to be "foreign agents" (in the style of Russian or US laws) could probably be used to further hinder the activities of foreign imams etc., although it's unclear what exact measures Macron has in mind for now, but it seems rather clear whom he has in mind (at least had in mind in late 2018):
French government spokesman Benjamin Griveaux told reporters last week that although they have not finalized reform plans, the training of Islamic clerics and their funding “are at the heart of the manner in which we are rethinking the relationship between the Republic and Islam.”
“Why is the question of funding of Islam central for us? Because today, we know that the funding comes from foreign countries, and it is not desirable to have a religion in France funded by foreign countries who in fact will be defending their interests. And so, it’s a political Islam,” he said.
Interestingly, Germany seems to think it has a similar problem. More recently Austria has expelled some Turkey-trained imams.