Issues before maiden voyage!!

Have you tried just using your hand instead of a hammer? Wrap the clutch actuator arm with a rag and press inward with a lot of force. Don't be afraid, shove it all the way in until it stops. Often, if the clutch pads are stuck, common problem with new engines, they will pop loose. If that doesn't work you won't hurt a thing by rapping on the end of the pin which protrudes from the middle of the engine drive sprocket with a hammer. Don't beat it to detah, just give it a good solid rap.
Make sure that the ball bearing is also there. It goes in the hole in the sprocket before the pin. These parts should be well lubricated with a good grade of bearing grease prior to riding.
Also assure that your clutch cable is adjusted correctly. You'll want little to no slack at the end where it attached to the clutch actuator arm with the handlebar lever out (not squeezed).
Let us know how you make out, please.
Tom

Correct but...don't expect it to ride like a normal pedal powered bike. There's going to be some parasitic drag from the chain/clutch/gears that you'll have to overcome. The only way to get away from that is to take the engine drive chain off.
You can pedal the bike but it won't be as easy as you might be used to with the drive chain on the sprockets. To overcome this you'll have to get that engine running right and then you can forget about pedaling Good luck.
Tom

If the sprocket is mounted where the disc brake rotor mounts then that's your problem. The sprocket is too far to the left. Mounting the sprocket to the rotor mounting area can be done but the sprocket should be machined to fit 'over' the hub; not on the outboard edge. A spacer must then be fabricated that will utilize the rotor mounting holes and the sprocket holes. I don't have a very good photo of this but I'll attach one that might explain how to do it.
Tom

Not really. There is very little space to move the sprocket outboard from the original position. The sprocket and the cover are part of the clutch actuating system and there is no space inside the cover to move/shim the sprocket to the left.

I'd like to see a photo of the sprocket you bought and had machined. The rear driven sprocket must be closer to the wheel/spokes to get anywhere near the correct chain alignment required. Offset to the left as I picture your sprocket to be, you're going to have problems.
Please try to post a photo/s of your rear wheel with the sprocket installed as you have it.
I have to wonder why you're opposed to mounting the sprocket with the kit supplied rag joint or one of the available sprocket adapters either of which would allow the chain to be correctly aligned with the engine drive sprocket.
Tom

Ramey,
Let's stop and back up to your original concern which as I understand it is "a clicking sound from the chain". Have you determined what that sound is, or in other words, what is the chain hitting?
From the photo it appears that the chain is very close to the seat stay. Is there any sign of interference there?
Another source of chain noise which is common with the Chinese 2 stroke engine is contact of the chain and the inside of the clutch actuator cover. There's a thread hear somewhere that shows where the chain hits and what can be done about it. I'll find it and post it later.
Back to your problem. I see you're using a chain tensioner. A good idea if some misalignment is necessary as it might be with your sprocket mounted outboard from where it should be. Try sighting down the upper chain path from the rear looking forward to the engine drive sprocket. Use a straight edge if possible to check how far off from center your chain is. Something like 1/4 to a maximum of 3/8" is tolerable. The tensioner will help guide the chain onto the rear sprocket. Anything exceeding 3/8" off from center and you need to do what is necessary to reduce it.
I hope all this helped and didn't confuse the issue.
By the way, have you by any chance contacted the seller of that sprocket to ask for advice? He might have a trick or two he could offer. I'd very very cautious about milling down the rotor mount area. One thing to keep in mind is that doing so will reduce the depth of the mounting fastener holes. That brings to mind another issue with this setup. You'll be driving the rear wheel with only six, 6mm screws. They'll need to be quality fasteners and kept tight. That's a lot of stress on those little fasteners.
Tom

EDIT: Here's that chain noise thread> http://motorbicycling.com/f30/motorized-bicycle-chain-noise-6682.html

Congratulations. Looks like a fun ride. Once on the road and the bugs are out, let us know how it rides. Certainly looks good. You're gonna love that disc brake out front.
Have fun but above all, ride safe.
I hope we have been of some help to you.
Tom