Another failure from Nick Compton and only a modicum of comparative success for Joe Root saw to it that more evidence is urgently required before either can be pencilled in for a debut as Andrew Strauss' replacement in the first Test against India.
Instead, by way of consolation but of less direct relevance to the grand plan perhaps, Bairstow (118) made his maiden first-class hundred in an England shirt. He and Morgan (76) shared a stand of 158 to rescue England from 66 for four on the way to 338 for six on day one of three against Mumbai A at the Dr DY Patil Sports Stadium.
Compton this morning followed his duck against India A on Tuesday with just a single, while Root succumbed to the last ball before lunch for a painstaking 28 on debut.
His was the third wicket to fall for just 10 runs against Mumbai A, leaving Morgan and Bairstow - runless between them - with an important job to do to ensure England's innings did not fall well short of expectations against a routine attack on a fair pitch. They performed admirably throughout a wicketless second session.
Morgan clubbed Suryakumar Yadav high over long-on for an early six in an 81-ball 50 also containing six fours.
Bairstow, scorer of 95 and 54 in England's most recent Test match at Lord's, may nonetheless soon find himself back in an understudy role again thanks to the 'reintegration' of Kevin Pietersen - the man he deputised for at HQ.
For good measure, he added a 158-ball century here, underpinned by sound shot selection and execution and completed with a flourish with a fierce cut for his 13th boundary off Kshemal Waingankar.
Even so, it was a fair bet Cook and coach Andy Flower would have readily swapped all of that for a more clear-cut indication of whether Root or Compton is likely to have what it takes in Ahmedabad on November 15.
The 'shoot-out' billed between them lasted little more than quarter-of-an-hour after stand-in captain Stuart Broad won the toss in a match minus the rested Pietersen.