Rules??
The rules that matter are: the Constitution.
The Constitution makes the VP, who is the President of the Senate, the SOLE AUTHORITY with Article Two POWER to conduct the election. The Senate has no Article One (or any other) power to do so, so what the Senators say doesn’t matter. What the President of the Senate says is all that matters.
If Pence opts to run the event under the Senate’s Rules, he’s free to do that. But, he doesn’t have to, and neither can the Senate’s Rules, nor any law passed by Congress, pretend to bind him... as his power is directed, plenary, narrowly focused, unique, and stated directly in the Constitution, Article TWO... which grants no authority to any other but the President, the Executive Branch, and those acting as the Executive in unique, specified circumstances.
The VP will conduct the election however he wants, and that includes that the VP will determine who runs the process in the event that he “steps away”... as it is his power alone to delegate.
If the VP decides that will be Grassley, then it will be. If the Senate’s Rules say it will be Grassley, then it will be, only as long as the VP opts to accept that. That the VP can accept the Senate’s rules... doesn’t REQUIRE him to do that.
And, while a change in the law adopted now wouldn’t matter even if they did try to pass a law over-riding the PLENARY POWER of the Executive, there is nothing in law now that I am aware of which even pretends it contradicts that.
Any change in law attempted now, would not apply to the current election anyway, but only future elections, given the unconstitutionality of ex post facto law. You can’t change the rules in the middle of an election, just because you don’t like how things are going as the election approaches... an issue that is already ripe for adjudication in the current election... on this, as well as on other grounds.