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All this electrical stuff has to have a good source of power but it must also have a good ground. Do you have a good ground strap from the engine to the airframe/battery?
I'm a little confused in what kind of starter you have. A "clicking" sound of a STARTER relay would indicate low current, but you've specifically stated you "pulled" the starter and your BATTERY relay clicked. So I'll assume you've still got your aircraft's original set up.
When you rewired your aircraft, did you accidentally mis-wire the battery relay? Look very carefully and trace this out. The battery POSitive post is connected to one side of the battery relay, and the OTHer side of the relay goes to the starter switch (which is activated by the pull cable.) Heres the important part: The small terminal of the battery relay goes to the aircraft instrument panel master switch, ...which GROUNDS that terminal to the airframe. That connection (of the small terminal to GROUND) is what activates the relay.
Mis-wiring that small terminal to another device lead (typically the generator/regulator circuit) may cause the relay to "click" closed (because it finds a ground thru that other device's circuits to the airframe) but that routing will not support a heavy current demand thru the relay.
Try disconnecting the small terminal's wire, and using a jumper wire, connect the battery relay small terminal directly to GROUND. (CAUTION: STAY AWAY FROM THE PROP!) This will activate the battery relay.
Then go to the cockpit and try an engine start. With all else fine, the engine should rotate nicely. Otherwise you have a bad/low battery, bad relay, bad starter, or incorrect/bad wiring.
(A common error when working with master switches is thinking that panel switch actually sends power directly to the aircraft systems. It doesn't do that directly. It does that by providing a ground to the battery relay small terminal. The other terminals (There are 4 of them) connect the voltage regulator's FieLD terminal to the generator's small Field terminal when the switch is activated. Accidentally reversing these two connections will give you the results you've indicated.)
NOte: After you've corrected this problem (if I've guessed correctly) you may find the engine starts but the generator won't charge. This may be because the gen/regulator has had it's polarity reversed in the above wiring error. All you should have to do is shut down the engine, but leave the aircraft master switch ON. Use a small jumper wire to momentarily jump across the BAT and the ARM terminals on your regulator to re-polarize the system. Start it back up and it should work fine.
The other possibility is that the generator charges,....and charges...and charges....(and your battery boils...and boils...and boils) ....because you've accidentally grounded the generator field windings along with the battery relay (instead of connecting it thru the master switch to the regulator FLD terminal.)