Yes, but "we want to use calculus to solve this" isn't a very natural constraint on the set of orderings. :) It's a "we want to make the math easier" constraint, not a "we have reason to believe that any rational agent should act this way" constraint.
Not that it's necessarily inappropriate in the example you give -- it probably makes sense there. Just a bit surprising that UDT would restrict itself in such a way.
In the UDT case, the set of outcomes is finite (well, or at least the set of equivalence classes of outcomes under the preference relation is finite) and the utility functions don't have any particular properties, so every possible preference relation the model can treat at all can be represented by a utility function!
(I should note that this is not UDT as such we're talking about here, but one particular formal way of implementing some of the ideas of UDT.)
A common mistake people make with utility functions is taking individual utility numbers as meaningful, and performing operations such as adding them or doubling them. But utility functions are only defined up to positive affine transformation.
Talking about "utils" seems like it would encourage this sort of mistake; it makes it sound like some sort of quantity of stuff, that can be meaningfully added, scaled, etc. Now the use of a unit -- "utils" -- instead of bare real numbers does remind us that the scale we've picked is arbitrary, but it doesn't remind us that the zero we've picked is also arbitrary, and encourages such illegal operations as addition and scaling. It suggests linear, not affine.
But there is a common everyday quantity which we ordinarily measure with an affine scale, and that's temperature. Now, in fact, temperatures really do have an absolute zero (and if you make sufficient use natural units, they have an absolute scale, as well), but generally we measure temperature with scales that were invented before that fact was recognized. And so while we may have Kelvins, we have "degrees Fahrenheit" or "degrees Celsius".
If you've used these scales long enough you recognize that it is meaningless to e.g. add things measured on these scales, or to multiply them by scalars. So I think it would be a helpful cognitive reminder to say something like "degrees utility" instead of "utils", to suggest an affine scale like we use for temperature, rather than a linear scale like we use for length or time or mass.
The analogy isn't entirely perfect, because as I've mentioned above, temperature actually can be measured on a linear scale (and with sufficient use of natural units, an absolute scale); but the point is just to prompt the right style of thinking, and in everyday life we usually think of temperature as an (ordered) affine thing, like utility.
As such I recommend saying "degrees utility" instead of "utils". If there is some other familiar quantity we also tend to use an affine scale for, perhaps an analogy with that could be used instead or as well.