Is it permissible not to include the grandfather among the heirs of the deceased in the legal declaration (i'lām shar'ī) due to his old age and lack of awareness, and to spare him the burden of official procedures, contenting oneself with praying for him, given that there is a family agreement on this?

Your uncle cannot waive your father's (your grandfather's) share of the inheritance. Waiving is considered akin to a gift, and a guardian is not permitted to gift anything from the assets of the ward. You must give your grandfather his full share. If he is of sound mind and understanding, he may dispose of it as he wishes. If he is not of sound mind, his share must be preserved for him and given to his guardian, and the guardian is not permitted to waive it.