I understand that in essence, your son is asking whether ἐρευνῶ can possibly be used together with a preposition (εἰς), or whether in this case the prepositional phrase is a bit dissociated from the verb.
Your son's question is good, but due to fact that the extant textual / inscriptional evidence from the ancient world is not complete, (and therefore the answer to such a question always needs to carry the caveat "as far as we can tell" or the like), I suggest that you also ask a related question.
If you want to use the εἰς apart from the verb, then the verb will have no object. So, you could ask, is the verb ἐρευνῶ is ever used intransitively (without a direct object).
Another approach is that perhaps you could consider the παρακύπτω in the following verse 1Peter 1:12 εἰς ἃ ἐπιθυμοῦσιν ἄγγελοι παρακύψαι., as you are considering your ἐρευνῶ. It might be part of Peter's idiolect (personal speech habit) to favour using the εἰς with verbs that have the same meaning.
"αἴκα" (The Spartan Ephors' reply to Philip II of Macedon) is even better than "nuts" (General Anthony McAuliffe reply to General Heinrich Freiherr von Lüttwitz).