Are there any Shariah-based impediments to inviting mourners to dinner to seek reward for the deceased and for the inviter, especially since it is accused of being an innovation (bid‘ah)?

The act of the deceased's family preparing a feast and inviting people to it is an innovation (bid'ah) that goes against the Sunnah. The Sunnah is rather that food should be prepared for the deceased's family and gifted to them, based on the saying of the Messenger of Allah, peace and blessings be upon him: "Prepare food for Ja'far's family, for indeed something has come to them that has preoccupied them."

Furthermore, gathering at the deceased's family's home is an innovation and a form of wailing (niyaha), based on the statement of Jarir ibn Abdullah al-Bajali, may Allah be pleased with him: "We used to consider gathering at the deceased's family's home and preparing food after burial to be a form of wailing."

Therefore, one should beware of these innovations and adhere to the Sunnah, and frequently seek forgiveness for the deceased and pray for mercy upon them, and give charity on their behalf if possible.