What would your approach be in a man currently on treatment for high-risk prostate cancer with ADT who does not have castrate levels of testosterone?

Yes, I would try alternative agents. If using Lupron, consider relugolix, degarelix, high-dose bicalutamide, or even adding an ARSI.

I would switch to a GnRH antagonist or add another agent.

It is perhaps worth noting that most (all?) randomized trials of ADT with RT combined agonists like leuprolide with bicalutamide. Ongoing trials (NRG-GU010, NRG-GU009) still require this. There is an interesting biological explanation for why the combination might be necessary for some individuals who have a genetic propensity for extragonadal testosterone synthesis. See, e.g., Hearn et al., PMID 29049492; Hearn et al., PMID 27575027. We f...

I would first measure testosterone using mass spectrometry. There's a fair probability in this case that testosterone is already at castrate levels.