My suggestions in bold, followed by the classical examples that inspired them.
madidus nasus / a wet nose
madidique infantia nasi / and the wet noses of a child (Juvenal,
fluens pituita / streaming snot
fluctus nasus / a streaming nose
nasus fluxit pituita / a nose streaming with snot
Pituita / slime, clammy moisture, phlegm is used to describe phlegm-like bodily discharges, often as one of the four humours. See, for example Cicero, Tusc. Disputations, 4.23 (bad blood, bile and phlegm) or Horace, Satires, 2.75 (an overabundance of intestinal “phlegm”).
But it is also used to describe a “viscous, gummy moisture that exudes from trees”, which sounds just like snot to me. Here Catullus uses it in exactly this way:
mucusque mala pituita nasi / mucus and bad slime from the nose
(Catullus, Poems, 23.17)
Fluere / to flow, stream, pour is used for rivers and streams, but also of bodily fluids, such as:
sweat: sudore fluentia multo bracchia (Ovid, Metamorphoses, 9.57)
tears: fluunt lacrimae (Ovid, Fasti, 2.820)
entrails: viscera lapsa fluunt (Ovid, Metamorphoses, 8.402)
soft brains (flowing through hollow nostrils like curdled milk flowing through a pliant oaken sieve – please see a doctor if this describes your runny nose!):
perque caves nares oculosque auresque cerebrum
molle fluit, veluti concretum vimine querno
lac solet …
(Ovid, Metamorphoses, 12.435-436)
proluvies nasi / an overflow of the nose
proluvies / an inundation, discharge, overflow. Used by Cicero (Letters to Quintus) to mean an actual flood but here are examples of bodily “floods”:
foedissima ventris proluvies / foulest discharge from their stomachs
(of the Harpies in Aeneid, 3.216-217)
mucosa ventris proluvies / a flood of mucus from the stomach
(Columella, On Agriculture, 6.7.2)
destillatio [nasi] / a dripping down [from the nose]
destillatio / a dripping down, distilling, running, from destillare / to drip or trickle down
The verb can be used of any liquid that trickles down (for example sap from a tree, see Pliny, Natural History, 23.72).
But here, Pliny uses it to describe the outpouring of mucus after a sneeze:
semen tritum et haustum naribus sternumenta movet et distillantiones
quoque capitis inlitum / the crushed seed inhaled by the nostrils
produces sneezing and also as a liniment [it produces] a flow [of
mucus] from the head (Pliny, Natural History, 20.123)
Destillare is used a lot by Celsus in On Medicine, for various types of dripping from the body: two examples are the drip, drip, drip of difficult-to-pass urine (2.7.13; 4.2.N); and watery eyes (6.7.A). However, he uses destillatio exclusively to mean a runny-nose cold:
tussis, destillatio … sternumentum / cough, runny nose … sneezing (2.8.22)
tussis, destillatio, raucitas / cough, runny nose, hoarseness (2.1.16)
Here, using destillare, he is describing the symptoms of a cold:
destillat autem de capite interdum in nares ... / moreover, there is dripping from the head sometimes into the nostrils ...
... si in nares destillavit, tenuis per has pituita profluit / if into the nostrils it has dripped, a thin phlegm flows forth from them
(note the use of pituita and also profluere – very close to your suggestion of perfluere. Profluere is also used in this sense of snot flowing from the nose by Columella, writing of sick cattle – pituita a naribus profluent 6.7.4.
So perhaps we could also use profluens pituita)
Likewise, Pliny also uses destillatio to mean a “runny-nose cold” – asperitatem fauciam et destillationes lenient cochleae / a sore throat and runny-nose cold are relieved with snails (Nat. Hist., 30.32).
Thus, of all the suggestions, I think destillatio [nasi] is probably the most accurate in so far as it is attested to in medical texts, speaking specifically of colds. However, it was enormous fun to track down all the other possibilities!
By the way, you mentioned rhinorrhoea, a Greek term, and so I was interested to read Celsus’ clarification of Greek terms for colds. First, he distinguishes between a cold where the nose runs and one where the nose is blocked. A blocked-nose cold he calls gravedo, and he notes that the Greeks call this κορυζας. The runny-nose or destillatio-type of cold, he says the Greeks call κατασταγμον (4.5.2).