Could the Senate legally hold an impeachment conviction vote in secret?
Under current rules, no.
RULES OF PROCEDURE AND PRACTICE IN THE SENATE WHEN SITTING ON IMPEACHMENT TRIALS
XIV. The Secretary of the Senate shall record the proceedings in cases of impeachment as in the case of legislative proceedings, and the same shall be reported in the same manner as the legislative proceedings of the Senate.
XX. At all times while the Senate is sitting upon the trial of an impeachment the doors of the Senate shall be kept open, unless the Senate shall direct the doors to be closed while deliberating upon its decisions. A motion to close the doors may be acted upon without objection, or, if objection is heard, the motion shall be voted on without debate by the yeas and nays, which shall be entered on the record.
XXIII. An article of impeachment shall not be divisible for the purpose of voting thereon at any time during the trial. Once voting has commenced on an article of impeachment, voting shall be continued until voting has been completed on all articles of impeachment unless the Senate adjourns for a period not to exceed one day or adjourns sine die. On the final question whether the impeachment is sustained, the yeas and nays shall be taken on each article of impeachment separately; and if the impeachment shall not, upon any of the articles presented, be sustained by the votes of two-thirds of the Members present, a judgment of acquittal shall be entered; but if the person impeached shall be convicted upon any such article by the votes of two-thirds of the Members present, the Senate shall proceed to the consideration of such other matters as may be determined to be appropriate prior to pronouncing judgment. Upon pronouncing judgement, a certified copy of such judgment shall be deposited in the office of the Secretary of State. A motion to reconsider the vote by which any article of impeachment is sustained or rejected shall not be in order.
Rule XIV requires the record be the same as other other proceedings.
Rule XX requires the proceeding be public.
Rule XXIII requires a roll call vote for each article.
For example, this, A resolution impeaching William Jefferson Clinton, President of the United States, for high crimes and misdemeanors., is a list of votes for the Senate.
I wonder if any other parts of the Constitution shed light on this issue?
Aticle I, Section 5:
Each House shall keep a Journal of its Proceedings, and from time to time publish the same, excepting such Parts as may in their Judgment require Secrecy; and the Yeas and Nays of the Members of either House on any question shall, at the Desire of one fifth of those Present, be entered on the Journal.
"[P]ublish", in this sense, means make public.