Is paying the guard to enter the Haram considered a grave sin because it is within the Sacred Mosque, and is the guard allowed to prevent entry?

The basic principle is that people should not be prevented from entering a mosque unless there is a clear harm. It would have been appropriate to maintain order and not shove the guards, but rather to look for another place. What you did by pushing the guard and entering forcefully was not proper, as it involved harming a Muslim and increasing the punishment for disobedience in the country and during the sacred month. Since you have repented, sought forgiveness, and regretted your actions, we ask Allah to accept your repentance, for "one who repents from a sin is like one who has no sin." And because the sin involves the right of a human being, it is obligatory to seek his forgiveness if possible; otherwise, seek forgiveness for him and supplicate for his well-being.