I think what you are referring to is the display of the Z axis tool position. There are 4 display coordinates on the standard Haas screen and if you go to the "posit" display and toggle the up and down arrows you can scroll thru the coordinate displays. Machine coordinates, Work Coordinate, Oper, and Dist to Go. You can also toggle thru these while running a program, or use the "curr cmds" screen which shows more info than just the program text, the newer machines display work and dist-to-go at the same time on the right in a separate display window.
The z axis position is always displayed as the machine coordinate plus the G54, G55, ...etc. active work offset applied. Depending on how you touch off tools this is usually the G53 absolute machine Z position. When G43 tool length offset is active the value stored in the offest table for that tool # is added to Z and the Z axis Work display shows the tool position that you expect to see. G43 is usually only active during actually running a program and is always cancelled when changing "modes" as you mention. I don't believe you can force G43 to stay active and I don't think you would want to do that for safety reasons.
Long answer, sorry but what you're used to on a "hobby" type cnc axis display position may seem counter-intuitive on the Haas, where a casual glance at the z axis position may not tell you intuitively where the Z axis is. When you press "tool offset measure" this enters a value into the tool offset table which is the difference between machine absolute Z zero and the location of Z when you push the button.
You'll find the "oper" position screen most useful for this since you can "zero" or put any value in the axis display coordinates and use this screen like a DRO and it doesn't change anything in your actual offsets tables. When in "oper" coordinate display page press the "origin" key and it will zero the selected axis. Type an axis letter plus a numeric value and press "origin" and it will change the axis display to the typed value. I think this is what you are looking for.