Not sure I duplicated your situation exactly, but I went into the Garmin 172, set the baro to 29.92 and let it climb to 3000. No problem. Then changed the baro to 30.00 and the plane slowly corrected back to the original 30.00 I set in. No problem.
Left the baro at 30.00 and set climb to 4000. It started the climb just fine. Then when it hit 4000, I noticed something. The screens inside showed it was at 4000 with the baro showing 30.00, but the external view was showing about 3915 and the baro incorrectly still showing 29.92. It appears the external view doesn’t change the baro reading, and shows the altitude as what it would be if it really were 29.92 rather than the 30.00 that I set inside the plane.
If I changed the baro back to 29.92, the plane move to the correct altitude that it was set to (4000) and it showed correctly both internal/external views.
So it looks like the external view isn’t changing baro reading, and thus incorrectly shows the altitude in that view.
Regards