Dick and Tim (and possibly Cass-don't know where that adoption stands) are true heirs whether or not Damian is his biological son, which I guess he would have to prove by stepping forward, claiming he's the son of Bruce Wayne, and taking a DNA test if possible. But Dick and Tim are Bruce's sons equal to him if he's Bruce's kid, and Bruce no doubt has a very specific will already.
But yeah, I've no idea how they're handling the whole "Bruce is alive/dead" thing. Tommy Eliot is I believe going to escape and start pretending to be Bruce, which he already has. I would assume the point of it is to spread out Bruce's death and Batman's as long as possible so people don't connect them dying at the same time (as Vicki Vale already has). Tommy was already posing as Bruce for a while so that kind of helps them there. Bruce has been seen alive after Batman was considered dead. Now Dick and Tim are just both claiming to have spoken to Bruce recently long distance.