Alford gives some thought to this question in his comments on Acts 16.20, which are worth considering.
Alford wrote:The distinction between ὑπάρχων and ὠν seems to be, that the former [ὑπάρχων] is used of something which the speaker or narrator wishes to put forward into notice, either as unknown to reader or hearer, or in some way to be marked by him for praise or blame: whereas the latter [ὠν] refers to facts known and recognized, and taken for granted by both. Thus, we may notice that, when the fact of Paul and Silas being Romans is announced to the jailor [16.37], it is not ἀνθ. Ῥωμαιους ὀντας, but ὑπαρχοντας; whereas here, both parties, the speakers and the addressed, being indisputably Romans, we have Ῥωμαιοις οὐσιν [16.21]. The account of this may be, that ὑπάρχω is predicated of something which the speaker informs the hearer, some prior knowledge which he possessed and now imparts,—εἰμι being predicated for the bare matter of fact. See [Acts17.27,29 21.20 (for both) 22.3; Gal 2.14 al] for ὑπάρχων: and for ὠν, [John 3.4 4.9a,b; Rom 5.10] al.
However, Alford gives a different explanation in his comments on Gal 2.14: Ἰουδαῖος ὑπάρχων "by birth, originally".
I reckon ὑπάρχειν frequently has the sense "to be inherently/really" (BDAG #2), but this has different pragmatic uses according to context.
It can be used to contradict a false impression or presupposition:
Luke 7.25 ἰδοὺ οἱ ἐν ἱματισμῷ ἐνδόξῳ καὶ τρυφῇ ὑπάρχοντες ἐν τοῖς βασιλείοις εἰσίν. [In case you thought otherwise], people who wear splendid apparel and live in luxury are normally found in royal palaces!
Acts 5.4 πραθὲν ἐν τῇ σῇ ἐξουσίᾳ ὑπῆρχεν After it was sold, the money was absolutely at your disposal [No room for disputing it.]
Acts 16.37 ἀνθρώπους Ῥωμαίους ὑπάρχοντας we who are in fact Roman citizens [contrary to what the magistrates supposed, see v. 38]
Acts 17.27 καίγε οὐ μακρὰν ἀπὸ ἑνὸς ἑκάστου ἡμῶν ὑπάρχοντα and yet in fact he (God) is not far from each one of us
Acts 19.40 μηδενὸς αἰτίου ὑπάρχοντος as there is in fact no reason for (today’s disturbance) [Emphasizes the fact of the situation in contrast to the evident presumption on the part of the crowd.]
It can also be used to stress the reality of a situation:
Luke 16.23 καὶ ἐν τῷ ᾅδῃ ἐπάρας τοὺς ὀφθαλμοὺς αὐτοῦ, ὑπάρχων ἐν βασάνοις In Hades he looked up, being in torment as he was [The rich man has not been portrayed as full of pity, so presumably ὑπάρχων is intended to confirm a reality that is already presupposed.]
Acts 2.30 προφήτης οὖν ὑπάρχων … προϊδὼν as he (David) was in fact a prophet … he saw into the future [Emphasizes David’s status as a prophet as something crucial for the interpretation of the psalm that Peter has quoted.]
Acts 27.12 ἀνευθέτου δὲ τοῦ λιμένος ὑπάρχοντος πρὸς παραχειμασίαν Since the harbour was in fact unsuitable for wintering in [Perhaps this is also in contrast to what might be expected from the name Fair Havens in v. 8.]
I suggest Acts 16.3 is similar. We have already been told in v. 1 that Timothy’s father was not a Jew, and this fact is emphasized as being common knowledge because of its importance in justifying Timothy’s circumcision.
ᾔδεισαν γὰρ ἅπαντες τὸν πατέρα αὐτοῦ ὅτι Ἕλλην ὑπῆρχεν. since they all knew that his father had in fact not been a Jew.
In the case of the Jews in the same verse (τοὺς Ἰουδαίους τοὺς ὄντας ἐν τοῖς τόποις ἐκείνοις), there is no requirement to imply anything special about their being in Lystra and Iconium, so the verb εἶναι is used. It's not in contrast with ὑπάρχειν, just the "plain vanilla" unmarked choice of vocabulary.
In Acts 8.16 μόνον δὲ βεβαπτισμένοι ὑπῆρχον εἰς τὸ ὄνομα τοῦ κυρίου Ἰησοῦ, the verb seems to contribute to marking the condition of the disciples as merely baptized into Jesus.
In Acts 19.30 δέον ἐστὶν ὑμᾶς κατεσταλμένους ὑπάρχειν καὶ μηδὲν προπετὲς πράσσειν, it seems unlikely after two hours of the crowd shouting their slogan that the city clerk is saying you should be restrained as you inherently are (unless this is diplomatic talk). He seems to be emphasizing the attitude he recommends in contrast to what he fears may happen : it is essential that you really be restrained and do nothing rash.