There is no gas-liquid transfer in the pH probe method, it's liquid-liquid across the DO membrane.

The membrane just maintains the separation of the KH solution,(non gas), the gases can move freely.

The accuracy of the color for the pH inn the drop checker is limited by your color perception, which is rather poor.

The pH meter is not.

So the pH meter method will give you much more accurate measure either way.
since the phase change is not required, and the volume of KH ref solution is small relative to surface area, the effect is rapid.

Note, the tip can have very little KH ref between it and the outside water, a thin layer is all that is required.

Regards,
Tom Barr