FWIW, I agree with Dash that the better place (assuming you can get there consistent with other responsibilities / priorities) is on the first base side. If there's no FPSR violation, it puts you in a better position for swipe tag / pulled foot and overthrow.

Had this discussion the other day in my pre-game. Both the Red book (PBUC as well as Jim Evans book) as well as the FEd have PU heading toward third on this play. The NCAA generally speaking has the PU heading up 1st base line for the pulled foot swipe tag/overthrow. Leaving the back door play at third to the BU. Basically in the NCAA, they're playing the odds on the play. That is, overall you will see more pulled foot swipe tags in a game than the back door play at third. So it comes down to what you feel is more important. The pulled foot swipe tag or the back door play at 3B, that also could wind up with a swipe tag between the ball, player and you. I prefer the Pro mechaninc, heading to 3B, UNTIl the play on R1 at 2B. If he is out, I cut across the diamond to the 1B line for the pulled foot, swipe tag help. If he's not out, I'll continue toward the library. If there is an overthrow, stop, reverse course and take the ball. You most likely won't be further than four to six steps up the line. Using this mechanic, as Jimmy say's , sometimes you gotta umpire. As the BU, just know you may have to do just that, move and gain an angle for the swipe tag. Another point often made is with two runners why are both of you with only one?

The Pro mechanic (as well as Evans) does not contemplate the FPSR, and NCAA has BU taking all plays on the bases, so PU never has the "back door" play at 3rd. Evans has PU heading towards 3rd in foul territory on DP ground balls up the middle.

With R1 and a ground ball hit, PU should be heading towards third in the event the ground ball gets through and there is a rotation. If the ball is fielded, PU stops and gets into position for continuing action at second on double play, third base side of mound, and subsequent assistance on pulled foot at first, usually a step or two towards first to create an angle.

With R1, R2 and bases loaded, and ground ball. Plate umpire hits the point of plate and adjusts to whatever play occurs. If a double play is attempted with first play at second, PU should get 1BLE to observe continuing action at second. He is then in position to see touches at home and possible pulled foot/swipe tag at first.

The FPSR responsibility falls on BU until the ball is released to first. Responsibility then shifts to the PU for the continuing action. There is no shared responsibility on the call at any time. It is either BU's, or PU's after throw.

If PU's initial position for the call is on the first base side of the mound, there is no way he can be in correct position in the event the ball gets through in a rotation situation.

The manual shows the PU umpire going up 3rd base side of the mound for a reason. Its probably the best place to be, no matter where the ball is hit. Yes, you can have potential swipe/pulled foot at first, but much greater chance of FPSR at 2nd. And if you hustle out when the ball is hit, you can get a good view of the play at 2nd. And of course any follow-up at 3rd, PU is right there.
Yes, your not going to get more than a couple steps for a look at first, but again, your primary responsibility is FPSR. Then get what you can at first.
Plus if things blow up at first, you may end up having a good look from where you are.
Also, if your going up the first base side, your overloading that side of the field.
Two-man is playing the percentages, and your much more likely to have issues with FPSR than you are swipe/pulled...

You have a much better look at the slide at 2nd from the 1B side of the mound. And under CCH mechanics PU has no play at 3rd on a ball that stays in the infield.