My Honeywell PRO 3000 thermostat is not activating the cooling system when prompted.

On the thermostat, the two temp readings are showing two different settings, one is the setting temp that you are selecting, and the other is the room temp.

yes, provided the same number of wires are used in both. Try programming thermostat to the same temp day and night All timme periods 7 days a week. this will turn it into a manual thermostat. Refer to Robertshaw manual for directions.

liquid crystal screen has defect.

try:
https://www.youtube.com/watch?v=0TzL_eLXEpk

or
https://www.manualslib.com/manual/502113/Honeywell-Pro-Th3000-Series.html

Lower temperature to lowest setting wait 1 1/2 to two min U should hear a click at thermostat & it should come on.

Right below the w setting on the base plate is the o/b connection. Put the O wire in this slot. During the initial test for heat and cool, the bias for cool for function 2 is 0 and for heat it is 1. To reach the programming press fan and Up arrow. Press next to reach function 2. Be aware the the O wire is energized and operates the reversing valve. Some manufacturers energizes the reversing valve to go into cooling, and some leave it deenergized to be in cooling. Select either 0 or 1 to make the system perform in the correct mode.
Hope this helps.

I subjest the honeywell pro 3000 is a very good stat it comes in programable and non programable their is also a pro 8000 that is great its touch screen and has many options but the 3000 is most cost effective

They will work, most zone controls only require the use of 1H/1C thermostats, the system stages and functions are controlled by the board itself.
http://customer.honeywell.com/TechLit/pdf/PackedLit/68-0237.pdf

the pro 300 is a single stage thermostat and can not be wired to control 2 heat stages seperately