The drop to 11.3v is an issue...lets figure out why you have that much drop.
If you are drawing more than 7 A out of a 7AH battery, it's going to drop voltage commensurate with the load applied.
Further, the battery should have 12.6 v after resting an hour after a charge.
An unloaded battery reading of 12.4 is only 75% charged. And an unloaded battery at 11.4 is beyond fully discharged. Probably going to need an off bike charger. Consider using a car battery to provide power for long probing investigations.
I recommend you disconnect the headlight while you are troubleshooting. That will eliminate a 3-5 amp load (depending on wattage). That's easy with the stock bike by just removing the headlight fuse. But, it seems certain that the bike builder didn't bother providing a schematic of his changes before making his profit.
The stock bike draws about 10 Amps with key switch and lighting on. Which is likely why the stock battery is 12 AH and can provide about 200 cold cranking amps for the e start. That 7AH battery is mostly for computer back up purposes and spends most of its normal life on float charge, only draining for power outages and allowing the computer to perform back ups and saving Data in progress before shutting down. It's a silly choice for an SOHC4 that doesn't provide enough alternator power at idle to keep a battery charged.
Good luck!