Honestly, what's wrong with following the instructions on the circuit page? That's what all those notes are for. They tell you how to adjust the circuit. While a capacitor will do the job, it can't simulate the weird inverse resistive load of a lead acid battery. Stick a battery on the circuit, connect up a voltmeter, and adjust the pot until the LED lights solid at the voltage you want the battery to charge to. The LED will begin to blink as you approach that voltage.
It looks like he has faults with the circuit he has built, and it might be far safer to ensure it's working properly first, before he connects a battery!
Can you explain the 'weird inverse resistive load of a lead acid battery'? I was taught that its equivalent circuit is a voltage source equal to its EMF, in series with its source resistance. Nothing inverse about that.