You combine it nicely, but it is difficult to do, apart from the problems that colleagues have already mentioned, remember about the condensation of moisture when you start blowing the air with an anemostat at 10 degrees Celsius, then water will start to condense on it. As if the supplied air is already dry because the moisture is left on the evaporator, but still water flows from the diffusers :)
If you connect the supply fan with an air conditioner with a capacity of 900 m3 / h in series, you will have the effect of such a sound at home as if you were sitting in an airplane on the wing, it is impossible to live in such conditions.
From my practice in home recuperation at a demanding user, 4m / s on the bus is max, and 2.5m / s in the branches to the rooms, preferably below 2
I agree with lukaszkot89 install splits in those rooms that you want to cool.
Once I did such an installation that the circulating air in the building was cooled, but it required a second ventilation system, running new ducts, etc. It works so that recuperation itself, and a separate air conditioning installation with an air flow 5 times greater than that of the hand.
Remember the comfortable air supply temperature from the air-conditioning is max 8K cooler than the temperature in the room, and the safe temperature is 5K