Don't worry about it.
I think it's only an issue if you have a Chinese name where the "middle"/2nd name is an important differentiator.

I booked a CX award using Alaska miles. Did not include my middle name on the ticket. Passport shows middle name.
Encountered the website warning during online check-in ("..does not match"). chose not to check in online. Completed check-in at airport at First class line.
Agent did not mention anything, no problems at all.

Extra detail: I have Western first & middle names with an Asian (non-Chinese) last name.