As 3 hour is more than sufficient, whatever excuse you use they will see right through it.
As a DM they will tell you SDC/SDS on the next flight. I don't think they can push it to the next day for free. They might waive the change fee but I think fare difference still applies.

You should either hope for IRROPS or SDC to the last flight out. I don't know do they still allow SDC to break the '4 hour' layover these days, but in the past I've done it several times in ATL.