I have scoured engraving settings with no luck in solving this— I’m getting hyphen intersection with lyrics at the beginning of a system that divides a word. The only thing that has avoided this issue is unchecking “Center first lyrics on system…” in Engraving Settings > Lyrics. I hate the alignment with that setting off, and I would think there’s some sort of spacing setting I’m just missing. Any suggestions?
Well, there is an option to remove them completely! 
Some combinations of hyphen settings – gap values, compression, etc – work better than others; and you may have to do a bit of trial and error.
You’re right that centring the syllables is more usual standard than what you have there.
I would have thought that centring the syllables would move the two middle lines away from the edge, giving the hyphen more room.
Honestly, I would activate the setting to remove these completely at the start of a line. No major hymnal publications use hyphens there, as far as I know. It was once a limitation of Dorico that was finally removed with version 3, IIRC.
The alignment settings in the OP are the same ones I prefer, albeit with the lyrics nudged a bit more to the right.
I cracked open a nearby hymnal (Lutheran ELW) and on the first page I saw there was an example, and the hyphen doesn’t appear on the continuation staff. So unless you are working with a publisher who insists upon it, I would remove that option. ::: Bill
Good to know that this is not standard for hymnals; I guess I’m so used to choral standards that I didn’t think to check!
Do you have a good method to nudge the lyrics to the right, or do you just do this as a manual override in Engrave Mode?
I do this manually in Engrave mode. There’s no automatic solution. In my opinion, good hymnal engraving requires nearly every syllable (and voice column!) to be nudged anyways, even with good global settings.
Good to know, thanks!