How does one say "stalker" in Latin? In the sense of, as the OED puts it, "one who follows or harasses someone (often a public figure) with whom he or she has become obsessed."
In Pro Fonteio, 20(44) Cicero uses two verbs in one sentence: of pursuing or threateningly advancing, instō, -āre, and pressing on or threatening by proximity, urgeō, urgēre, ursi (infero is also related here, but wouldn't be specific enough out of context):
infestis prope signis inferuntur Galli in M. Fonteium et instant atque urgent summo cum studio, summa cum audacia.
Literal: “Carrying unequivocally hostile standards, Gauls [push/attack/advance on] M. Fonteius and [threaten/advancing on] him and [pursue/follow on heels of] him with the most zeal, and the most audacity [possible/imaginable].”
[[ commentary: infestis … signis is an Abl. Abs.; inferuntur is not passive but rather medial in this context, which had reified in many daughter IE languages into a single mediopassive verb aspect; the current thinking is that PIE did not have a past tense proper, exactly like the most ancient Hittite and the most modern Russian don't, and got by the aspectual use of verb form, as we see here. ]]
My feeling is, if you're looking for a classical word for your translation, the verb instō is the best. It does not have an attested cognate noun, however. It's stem is inste- (despite PERF institi), because of its internal form (in- + ste-), with the stem sound change caused by the prefix (the process long ended and perceived as irregular by the Classical period). But neither *instetor nor *instator are attested as nouns.
If you go the former route, and really need a substantive (which I believe to rarely be an unavoidable case), consider nominalising the PresAP, instans, in your context. To me, it sounds the most fitting semantics for the the period the language was spoken by the highly-educated people.
Note that both the two verbs are wanting attested PP form. urgeo is also somewhat irregular. This cripples your chances of nominalising the verb w/o a ridiculous result.
TL;DR: Try to use the verb instō if you can do it at all, as Latin likes verbals much more than English (strung substantives like "End county freeway speed limit" is not good Latin); as the second choice, if substantiation is unavoidable, then a substantiated in context PresAP instans, which is still a verb form, although non-finite, is your second best choice.
The verb "to stalk" has its origins in the world of hunting, "to stalk one's prey": the person or animal who secretes self, creeping silently behind cover, in pursuit of the target. It is easy to see how "stalking" acquired its contemporary, insidious connotation.
There may not be an exact equivalent, in Latin, when "stalking" (contemporary) would not have been seen as a crime.
There is e.g. "venator" = "hunter". A better fit may be "praedator", OLD, (masculine noun) = "prowler" (English-to-Latin section); in (Latin-to-English) plunderer, pillager; hunter. The "prowler" may have been the progenitor of "the stalker" but, although "praedator", reminiscent of the English "predator", sounds most appropriate it has more do do with thievery, in Latin, than the sexual perversion of an inadequate and/ or the malicious intent of an ex-partner. The "prowler", if memory serves could also be a night-time burglar/ thief, which parallels the Latin.
Interesting: I have not heard the term "prowler" for a long time. An old Oxford (English) dictionary (1979) does not list the contemporary meaning of "stalk". The nearest listing: "Pursue (enemy, game) stealthily".
Whereas "praedator" may not go far enough, another term, which has resonance in English, may go too far: "pestis" (feminine noun) = "pest" (English-to-Latin). Supplementary meanings (Latin-to-English) = plague, pestilence; destruction, ruin, death.
In English, "pest" can mean anything from a naughty child to plagues of locusts devouring crops. It would be a mild term for a stalker. The latin, more apocalyptical, "ruin/ death", at least makes the point that sex-pests can be dangerous and sometimes kill their victims.