I feel like forcing the unit to cast it's spells would be a nightmare to deal with depending on how the map works. For starters, if you trigger any of your spells then they would need to be coded so that they aren't going to cause problems. Second, you have to consider the targeting type for the ability. There's Instant, Target Unit, Target Point, and Channeling that all need to be considered. On top of that, if the ability targets a unit then you would need to consider the targeting filter, does it target Allies, Enemies, Air, Ground, etc...?
Plus what Pyro said about the animation backswing/castpoint. However, once we can manipulate this with triggers you could temporarily set these values to 0 before casting the spell and then reset them back to their default values afterwards.
I suppose if you really want the cooldowns to be displayed then this is the best option available, but otherwise just disabling the ability and then re-enabling it would probably be a lot easier to do. Although, if he wants the abilities to be disabled for different durations (like based on their cooldowns) then it would get slightly more complex but still very doable.