The chord progression in the first two verses goes G D Em G7 C Cm G. Obviously, just using my little finger to fret the high E string at the third fret to change from the regular C shape is a lot easier than going to a barre chord to get a Cm, and to my ear it sounds just fine.
Cool.
If you have Griff's theory course, this is
the progression he refers to in Lesson 20, "The Beatles iv Chord".
That's what he calls the move from a IV chord to a iv. In this case, C Major to C minor before resolving to G.
Problem with doing this from an open position C chord, as you say, is there's no "easy" way to move from an E to an Eb in this position.
As you've "discovered", staying on C Major still works, however, leaving the Eb out doesn't give the sense of movement from Major to minor.
You could use the Cm barre chord shape
without the barre, just using your 1st finger on that 3rd fret G. Fingers---->XX3421
Or try a Cm7, which is like the C7 with the 1st string G. But you still gotta use a barre to move down to that Eb.
Just a couple of options to get that Eb in there somehow (
if you want to).