Rather strange reason for failure. Usually they complain when you've
done the reset recently and they think you are hiding a problem.

I general, the check engine should light when you get a code in the
computer but I think it does watch for multiple occurrences before it
lights the dash light.

If there are Autozone stores near you, go to one. Ask them to read the
codes out of the computer and tell you what they see. It should match
the above list (mostly). Then have them reset the computer. They will
do all that for free.

I don't know how re-inspection is (if you pay for it, etc) but I'd try
for a re-inspection. The only issue I see is that, if they are like
other states, they'll complain about the reset unless you drive a lot
of miles.

Lastly, I'd guess you actually have a problem with that car. You may
or may note get through inspection but you should get it fixed. My
guess would be a bad O2 sensor, they are not uncommon failures. If
it's been gone a long time you may have cat converter issues. You'll
need to do some testing/work or have a shop do it for you.