Forget nonconservative fields. Put your rod in a conservative field, an electrostatic field. It will 'charge up' by induction. You will have opposite charge at the extremes. Now measure your absolute voltage with your method. It should be zero.
Is it zero?
It should measure zero, simply because it needs to. I have not come up with an elegant, direct proof, but I think I have one that is easy enough to understand.

I realized the field lines may not be exactly as I've drawn, but whatever they are, if I provide a path from each charged end out of the field along a path that is always perpendicular to the field lines and then connect them to a voltmeter and resistor, I can see if a current flows in the resistor. If it does, I can then extract energy from a static, conservative and irrotational E-field, which violates conservation of energy, among other laws. IOW, it just isn't done!
I can do the same with my brass balls and I would expect the same result. It takes no work to move them on a path that only cuts the field lines at right angles, so if they get charged up in the process, I could do it repeatedly and again, extract energy from a system that we know we can't get energy from. Another way of looking at it is that the charges will have no force causing them to go from one point to another because the fields are balanced, (E
ind + E
coul = 0) but I'm unable to prove that more directly, especially in the universal case as you demanded.
Now in the case of the straight wire tangent to a rotational, non-conservative field, things change. Specifically, we know experimentally that we
can take conductors (or brass balls, but I haven't proven that experimentally) out of the field and then extract energy from the system as long as the rotational E-field continues. We keep hearing that things are different in a non-conservative field, well that is one of them. If you look at my drawing in my last reply to
bsfeechannel, you'll see what I mean--the loop is not 'closed' until you are far enough away that the field is negligible, so the act of closing it only allows current to flow. If you had the ideal voltmeter and thus did not close the loop, you would still see the voltage.
In conclusion I guess I'm saying that the rules are different for conductors in static, irrotational, conservative fields than they are in dynamic, rotational, non-conservative ones--which we all knew already, right?

Exactly how that manifests itself is something that perhaps someone else can expound on, but the basic facts have been shown experimentally. The rule that a conductor is an equipotential in a the static, irrotational, conservative field may not apply here and adopting the 'integral loop' rule seems to me to be a crutch for a failed model. The rotational E-field exerts its force locally and tangentially and the rest of the story is just the process of adding it all up. The 'closed loop' requirement in general is a red herring and demonstrably false--I could, in theory, use multiple torii and
construct a linear accelerator by lining up the relatively straight inner sections of the rotational E-field for as long as I like. Sorry, the drawing isn't very good....
Edit: "Linear accelerator" is a flawed statement because I had a conductor in mind, not a loose charged particle. I'll try to refine what I'm trying to say when I get time. But the point will be that I can develop an observable, usable and perhaps measurable phenomenon at the ends of the straight conductor without ever contemplating any loop at all, something that can't be done in a static, irrotational conservative field.