Could the single Senator who opens a pro forma choose not to ask himself (or herself, as the case may be) whether there's a quorum and pass a bill 1-0?
No. Any such bill would not meet the necessary steps required for passage. Bills must be "read" on legislative days and pro forma sessions do not count as legislative days.
Rule XIV 2.
Every bill and joint resolution shall receive three readings previous to its passage which readings on demand of any Senator shall be on three different legislative days, and the Presiding Officer shall give notice at each reading whether it be the first, second, or third: Provided,That each reading may be by title only, unless the Senate in any case shall otherwise order.
Further to the point:
The Chair has stated in response to an inquiry that a pro forma session does not count as a legislative day for the purpose of advancing legislation under Rule XIV.
As a point of clarification, the Palm Sunday Compromise was not during a pro forma session; but rather a conditional adjournment. (S.Con.Res.23 - A concurrent resolution providing for a conditional adjournment or recess of the Senate, and a conditional adjournment of the House of Representatives.)
As of this writing (August 21, 2020), the Senate is meeting in pro forma sessions under an order issued August 13, 2020. The order specifically excludes the conduct of business.
Congressional Record - Senate
ORDERS FOR FRIDAY, AUGUST 14 THROUGH TUESDAY, SEPTEMBER 8, 2020
Mr. MCCONNELL. Mr. President, I ask unanimous consent that when the Senate completes its business today, it adjourn to then convene for pro forma sessions only, with no business being conducted on the following dates and times, and that following each pro forma session, the Senate adjourn until the next pro forma session: Friday, August 14, at 10 a.m.; Tuesday, August 18, at 12 p.m.; Friday, August 21, at 11:15 a.m.; Tuesday, August 25, at 2:30 p.m.; Friday, August 28, at 2 p.m.; Tuesday, September 1, at 7 a.m.; Friday, September 4, at 10 a.m. [Emboldening added.]