It will be under carbonated. You have to realize that the pressure of the CO2 in the liquid needs to equalize with the pressure of the CO2 in the headspace.
Mixing the syrup with club soda in the keg will work, but the agitation from the mixing process will bring a lot of CO2 out of solution.
If you kept everything super cold and sealed the keg before agitating, then you may be able to keep enough CO2 in solution for it to be acceptable. But you still face the problem of losing pressure as you serve.
If you go through it quickly for a party, you might be ok, but if it sits any appreciable time between pouring a glass here and there, then your root beer will eventually become under carbed.