Finding practical solutions for discharging and charging a capacitor
Can I charge and discharge at the same time?

No, current only flows one way. The remaining current would need to come from the charger. So if 2A was going to the caps and the load needed 1A, the source to charge the caps would need to source 3A.

I see that every 1 second my voltage from the capacitor drops 0.05V, so when I get to 1 second I have 2500mW of power left. If my load needs 1700mW, do I subtract that and I'm left with 800mW of power to deliver to my system while I charge my cap?

No, because you need to deal with energy in joules. Watts is joules per second, if you used 2500mW of power for 1 second it would be 2500mJ, that is what needs to be subtracted from the capacitor.