As you've mentioned, the 12th amendment to the Constitution, enshrined in statute in 3 U.S.C. §15 grants the role of presiding over the counting of the electoral votes in Congress to the President of the Senate, not explicitly the Vice President. As a result, the same provisions apply as would in any other circumstances when the VP is absent or unable to attend; as defined by the Constitution:
Article I, Section 3, Clause 5:
The Senate shall chuse their other Officers, and also a President pro
tempore, in the Absence of the Vice President, or when he shall
exercise the Office of President of the United States.
This would not be without precedent; although an absence, rather than a death. After the 1968 election, Vice President Hubert Humphrey was absent at the counting of the vote. The process was overseen instead by Richard Russell, a Senator from Georgia.
The House of Representatives' website has a nice article on this:
When Congress gathered to count the Electoral College in a Joint
Meeting on January 6, 1969, Vice President Humphrey was overseas
attending the funeral of Trygve Lie, the United Nation’s first
secretary general. In Humphrey’s absence, Senator Richard Russell of
Georgia, the president pro tempore of the Senate, oversaw the
proceedings. Two mahogany boxes containing the Electoral College
certificates from all 50 states and the District of Columbia sat in
front of him on the House rostrum.
Source: Collection of the U.S. House of Representatives
As President pro tempore of the Senate, Senator Richard Russell announced Richard M. Nixon the winner of the 1968 presidential election, after the House and Senate debated the objection of a North Carolina elector’s vote.