Check-in people (at least ICN returning to USA) have to "correct" and sign/stamp your boarding pass if your passport name does not match the itinerary name. Anyone else further down the path could give you trouble for any reason.

I made 4 roundtrips in 9 months with this hassle (plus not properly accruing miles, dealing with mileage claims long after the flight, etc) before I finally submitted the form to get it fixed (at SFO, upon completion of my trip). Upon arrival, I just had to go back to the departures area and drop it off at the Skypass counter. They updated my profile immediately.

My next 3 roundtrips in 10 months no longer had the hassle, and I utilized Morning Calm status, although way later I reissued an old ticket with the old name, no status/membership-number, and actually am delaying claiming the miles because I don't want my status to renew too soon. Probably the last part of this ordeal of a simple mistake.

Just get the Skypass part updated/fixed. It takes a few minutes once you're at the airport - not sure otherwise.
You probably noticed that the award ticket forces you to use the Skypass name, so try to avoid locking yourself into the "wrong" name if possible.

On the Chase side, I think it's fine if Chase has a slight mismatch, but I don't know for sure. Passport+Skypass should be an exact match.