As far as the politics qua media politics, Bush could pick Mario Cuomo and still get hammered. No pick is a good pick for a left-leaning media covering a GOP president. Crony, schmony. JFK made his brother the AG and nobody in the press worried their little pea brains about it. So f*ck the media.
As for the conservatives, they're equally hard to please. Nominate anybody, I mean anybody, and you'll get a certain segment of the conservative movement pissed at you. So f*ck the base.
From a coldly calculated perspective, it comes down to this: As you noted, Bush doesn't have to face his base at the polls again. (He has no heir apparent to protect, either.) But he does have to face Congress every day for the rest of his term. So, to recap, f*ck the base and f*ck the media too. How much goodwill this will buy from Harry Reid is an open question, but that's the goodwill he needs. If the Dems turn around and filibuster a compromise candidate, it just proves them to be the moribund, useless sack of horses asses we already suspected them to be.