Would wiring the constant power directly to the battery fix the problem?

Mine is already this way.

Perhaps with a smallish capacitor inline?

Once I had a capacitor network inline for my Mk1 (and it's not as simple as just wiring one capacitor, you also need a diode and the correct wiring and grounding), and this prevented some of the problems on the Mk1. But a properly-wired Mk2 shouldn't need a capactor. Heck, the Mk1 shouldn't have needed it either, but I deliberately wired the Mk1 incorrectly for my own reasons.

We've both got Hondas, so perhaps this is common with how honda's constant power in the dash works.

It's got nothing to do with the dash wiring. It's that Hondas have high-RPM starter motors that really draw a whole lot of juice from the whole car's electrical system.