What is the title of the person who is supposed to find out how many votes the party can rely on?In the Senate Republican Caucus
Officially, or formally, the title of the person you are thinking of is the "whip." In this case it would be the minority whip because the Republicans are in the minority in the Senate. This post is currently held by Jon Kyl (R-Arizona).
In reality, any member of the Senate who wants to can do this. If the leadership of the party wants, it can have someone who's really involved with that particular bill do the counting.
But the person who is officially in charge of counting up who's voting for what and trying to make sure the members vote in the "right" way is the whip.
The title comes from a fox hunting term "whipper in" which is a person who rides around making sure the pack of dogs doesn't split up and go separate ways.