The big battery would work with this circuit, but its overkill just for crank survival. The one on the circuit diagram is only 0.8Ah...and a 2.8Ah battery should be more than enough to survive a crank for most PC.
The limiting resistor needs to be recalculated to a lower value and a higher wattage otherwise your large battery will take forever to charge up. The diodes need looking in to as well, the one I stated above can only handle up to 3A. Getting a smaller battery is not that expensive, less hassle in installation.
As I said before that circuit was designed for low powered equipment so you cant just use those components I listed. Its main function is surviving the crank and not for extending the PC running power...run your engine if you want this until you run out of fuel
If you do want to use the second battery as an extra running power for your PC then another approach might be a better solutions.
OPPS I missed your last sentence...
You still need to upgrade your diode to handle more than 3A. A 5A diode should be OK for your application. Find a diode with the lowest voltage drop on the link I posted above