You will cool the air down but not enough to eliminate water in the tank. I have 4 - 8' tall, upright 3/4" L type copper columns with 4 vertical water traps downstream of the tank and an additional 10' long 3/4" L type copper run to a desiccant tower and moisture bottle. It uses the mass of the copper pipe and the superior heat transfer to cool the air enough to remove the water. The time it takes to travel 40 plus feet through the copper piping is what allows for effective cooling, water separation and removal.
I'm not saying what you are planning wont be somewhat effective but you should plan on a regular tank draining whether it be an auto drain or a simple manual valve, just the same.
My compressor is in a shaded, cool, insulated outdoor enclosure behind the wall in the first pic. It has air from the climate controlled garage vented into the enclosure.
The second pic may be a bit confusing having two hose reels, one being oil less and used only for blasting and painting. This system was built in less than a day and is very effective at removing moisture but I still need to drain my tank regularly.
I know you are attempting to nip the problem in the bud by cooling the air before the tank but I believe it will not remove as much water as you think it will. Good luck with it.