Is it permissible to supplicate against a head of state in the mosque if that would cause discord and problems? And how should a Muslim supplicate concerning this matter?

It is permissible to supplicate against an oppressor. Indeed, it is authentically reported that the Prophet, peace and blessings be upon him, supplicated for the salvation of the oppressed and against their oppressors. He also supplicated against rulers who caused hardship to the Ummah. However, if supplicating against a specific oppressor leads to a greater harm, then it is not permissible to supplicate against them at that time. Instead, one should supplicate generally against oppressors, which would then include that specific oppressor. This is because warding off harms takes precedence over acquiring benefits.