If I were the President, I would most certainly file such a case.
The Senate may choose to assign a committee to vet the nominee and perhaps even provide a recommendation, but it is the right, I would argue, of each and every Senator to vote on a Presidential nominee a President nominates.
There's got to be a reason Bush isn't filing such a suit.
IMHO he should get going and establish the principle before he needs to make a SCOTUS appointment.