I take full responsibility for the accident. However, he doesn’t want to go through insurance for financial reasons. His wife is dealing with early onset Alzheimer’s, so I don’t want him to be without a car while I get it repaired. Can this be fixed with the right parts?
Former repair estimator and Honda fan here 
To be honest, based on the photo, it looks like it just needs a new grille. However, Honda says whenever the grille is removed or replaced, the front radar (lower square object) must be calibrated. Calibration can cost anywhere from $300 to $1500. Can you skip the calibration? Sure. Could it malfunction without it? Yep.
You should decide what risks you’re willing to take. My personal opinion: even though I get the hesitation with insurance, you should still consider calling your insurance.
@Micah
The radar isn’t attached to the grille at all. If you don’t mess with it during the grille replacement, there’s no technical reason to recalibrate. Honda says it needs recalibration because they’re the only ones that can do it.
@Houston
Thanks for the correction! After looking into it more, you’re right. You can technically remove the front bumper/grille without affecting the radar mount, which means no recalibration needed. Just so you know, Honda doesn’t lock down their radar software, so you can actually get it calibrated by third-party vendors, who are usually a lot cheaper than dealerships. If you ever need calibration, I’d recommend calling around to local options.
Father-in-law’s, am I right?
I grabbed that grille piece off eBay for $37. It came with chrome accents instead of black, though.
Just because he doesn’t want to go through insurance doesn’t mean you can’t report it to yours. You’re at fault, so you can file a claim with your insurance… assuming you have the right coverage.