The early days of 'what to do?'
Idiomatic use of the expression "What to do?" (or "What to do!") as a sort of rhetorical or musing question to oneself goes back a number of decades in English. Its origin may be in a longer expression of the type "asked X what to do," as in this deposition of December 24, 1703, by Colin Campbell from Journals of the House of Lords, volume 17 (1704):
I met Twice with Mr. Ferguson in The Vine Tavern in Holbourn, where Mr. Clarke was present. I told Ferguson, "That I had been examined by the Earl of Nottingham:" I asked his Advice what to do. He told me, "Certainly I should be re-examined, and put into Custody."
The sense of the relevant sentence here is "I asked his advice as to what I ought to do." This is consistent with WS2's suggestion in a comment above that "What to do?" amounts to a shortened version of the question "What am I [or are we] to do?"
The expression appears in a different sense as a freestanding question in Francis Beaumont & John Fletcher, A King and No King (1619):
Bessus [a captain]. And then, if please your Majesty to remember, at on time, by my troth I wisht myself wi' you.
Mardonius [a captain]. By my troth thou wouldst ha' stunk 'em both out o' th' Lists.
Arbaces [King of Iberia]. What to do?
Bessus. To put your Majesty in mind of an occasion; you lay thus, and Tigranes [King of Armenia] falsified a blow at your Leg, which you by doing thus avoided ; but if you had whip'd up your Leg thus, and reach'd him on the Ear, you had made the Blood-Royal run down his Head.
And again in "The Trial of John Tasborough and Anne Price, at the Kings-Bench, for Subornation of Perjury, February 3, 1679," in A Complete Collection of State-trials, and Proceedings for High-Treason, and Other Crimes and Misdemeanours, volume 2 (1730):
Mr. Serj[eant] Maynard. ... Now after Trials for so high a Crime in the most publick Way, here come Price and Tasborough; What to do? She indeed before, but both Tasborough and she afterwards, by Rewards, and such Temptations, endeavours to disgrace his Evidence; but there is not only that, but the great thing looked after by their Party, was, the Issue of it; upon this all the King's Evidence were to be hang'd.
In both of these instances "What to do?" might be translated as "To what purpose?" or "For what reason?" In the perjury trial example, it appears to be to be short for "What do they come [or intend] to do?"
The expression also appears in a journal entry by John Wesley of May 26, 1778 in Works of the Rev. John Wesley, volume 1 (1809):
Friday 26, My soul continued in peace, but yet in heaviness, because of manifold temptations. I asked Mr. Telchig, the Moravian, what to do? He said, you must not fight with them, as you did before, but flee from them, the moment they appear, and take shelter in the wounds of Jesus.
Here the sense of the expression appears, once more, to be "what was I to do?"
'What to do?' in recent times
To gauge the popularity of rhetorical use of "what to do?" I ran a Google Book search for instances where the phase is repeated consecutively ("What to do? What to do?"). The search returned more than 200 matches—the vast majority of whch were from books published within the past 25 years.
One exception is from San Francisco Classroom Teachers Journal, volumes 18–21 (by 1934) [combined snippets]:
WHAT TO DO! WHAT TO DO!
HAVE A DOCTOR AT ONCE if you have any sickness, or accident No doctors are specified, but disability begins FROM THE DATE OF THE FIRST MEDICAL ATTENDANCE.
Another relatively early instance occurs in Sam Bobrick, Hamlet II (Better Than the Original) (1985):
HAMLET. To be or not to be. That is the question. Do I get involved in this family domestic drivel, which could go on forever, or do I end it all by killing myself, which is kind of stupid when you realize someday all of Denmark will be mine—which actually is another good reason eor killing myself. But then if you kill yourself, how do you know things are any better on the other side? Suppose there's no Chinese food? What to do? What to do?
I have vivid memories of playing Canasta with my grandmother (who grew up in Ontario in the late 19800s/early 1900s) in the 1960s and 1970s. One of her favorite parts of the game was agonizing over which card to discard at the end of a turn, and one of her favorite expressions—along with "Oh, forever more!" and "What a fistful!"—at such moments was "What to do? What to do?"
Your manager may be correct in suggesting that "What to do now?" is less completely formed than, for example, "What do we do now?" is. But it is surely a widely recognized idiomatic expression in English—and it has the potentially significant advantage of inviting hearers to decide for themselves whether you have in mind "What am I to do now?" (and thus taking on yourself the full burden of responsibility for the next step) or "What are we to do no?" (and thus encouraging others present to accept some of that burden, too).
The phrase also seems to be growing in popularity in recent decades, at least in published writing.