I would class the following as reasonably innocuous in tone but only because of the works they appear in. That is to say, there is no reason for their authors to be scurrilous or vulgar.
- lotium facere, see Cato, On Agriculture, CXXVII
- lotium ire, see Cato, On Ag., CLVI
- urinam facere, see Pliny, Natural History, XXVIII.LX; also Columella, On Agriculture, VI.XXX
- urinam reddere, see Pliny, Nat. Hist., XX.XXIII & VIII.LXVI & VIII.LVI; Celsus, On Medicine, 2.14
- cupiditas urinae, in Celsus, On Med., 2.12
- urinam fundere, in Pliny, Nat. Hist., XXIV.CXI (but note that this is describing dogs)
- exinanita urina, in Pliny, VIII.LVI (used of hedgehogs)
exonerata vesica occurs in the Satyricon (27) and Macrobius has ire minctum in the Saturnalia (3), both of which sound reasonably benign to me but the authors make me think we should tread with caution.
In truth, it's difficult to clearly define register and tone, possibly because the register was narrower and/or more flexible than our own. Lotium is a case in point. When Cato uses it in On Agriculture, it seems quite straightforward and polite. It sounds less polite when Catullus uses it to describe a Spaniard brushing his teeth with it (XXXIX.21). It sounds positively vulgar when Petronius uses it to describe someone who non valet lotium suum (Satyricon, 57).
Similarly with both mingere and meiere. Although they occur across a broad spectrum of genres and authors they are, nevertheless, used liberally by authors of invective and satire, such as Catullus, Juvenal, and especially Martial. Indeed, Martial's use of meiere seems deliberately to play on the ambiguity of its meanings (to ejaculate and to urinate); 46.2, for example, is often translated "to pee" (or similar) but is open for interpretation. Either way, it's horrendously obscene.
On the other hand, Catullus uses meiere in XCVII to reference a female mule and so can only mean urinating rather than ejaculating. But, in the context of the poem, Catullus probably means something more like "pissing". All of which is to say that meiere, in the hands of some authors at least, is clearly meant to be offensive in the extreme.