Is the Imam's prayer over the deceased outside the mosque with a loudspeaker, while leaving the worshippers inside, considered an innovation, especially since the Prophet, peace and blessings be upon him, used to pray over the deceased inside the mosque?

The Prophet, peace and blessings be upon him, mostly prayed over the deceased in the musalla (prayer area), and occasionally prayed over them in the mosque without any dislike. The Imam's use of a loudspeaker is a good practice to achieve the objectives of Sharia (definition) and for the congregants to hear his voice. There is no harm in the Imam praying in the mosque's courtyard if he is praying in front of the congregants and they can follow him. If there is a barrier that does not prevent following, then there is no issue. It is preferable that they pray with him in the place of the janazah (definition) and line up behind him.