That’s good thinking, and it’d be silly not to address it, but it didn’t happen. Jake didn’t watch the whole thing. Prismo didn’t turn the TV on until Finn had already put on the crown, frozen everything, AND set off the bomb.
The Destiny Gang says ‘Mertens’ one more time after this, but that doesn’t matter, either. Because before we even fully rejoin Finn [as the episode switches back and forth between Finn in Farm world and Jake in the time room], Prismo mutes the TV.
And he keeps it muted, beyond the final Mertens mention, while he, Jake, and the Cosmic Owl hang out, ignoring the TV.
The TV is only finally unmuted accidentally by the Cosmic Owl after Farm world Jake fell into the radiation and was mutating into a Lich. You’ll recall that this itself is the impetus for Jake making his wish and ending our glimpse at Farm world. Jake learned very little of anything from Farm world, and he did not learn Farm world Finn’s last name.
Another possibility we were posed is that Joshua and Margaret found Finn when he was old enough to talk, and he told them his name himself. Again, it’s tricky to rely on Finn’s own memories of his infancy, but in the season 1 flashback, Joshua expressly mentions that Finn doesn’t talk.
So maybe. I don’t think that’s what happened, but Finn’s memories may be unreliable, and I won’t rule it out.