Q: Is it permissible, when a Muslim who is Fasiq dies, to ask Allah to have mercy upon him?
A: Yes, it is permissible to pray for Allah's Mercy be upon him and supplicate to Allah to forgive him. Likewise, Funeral Prayer should be performed over him if he is a Fasiq, not a Kafir (disbeliever/nonMuslim).
Allah is the One Whose help is sought.