Toothed "alligator clips" will fill with crud and be difficult to clean. For normal Model Rockets you should use smooth jawed micro clips. There are none sold at Radio Shack that are any good. The last time I looked they still only sold the steel or copper coated steel. The copper coating will wear off almost instantly and they will rust.
You want either the solid copper Mueller #34C micro clips or the stainless steel version sold by NARTS. Copper is easy to solder to, but stainless requires a special flux that contains hydrochloric acid. If you do not want to buy a bottle of the liquid flux from a welding supply house, you can simply borrow a small amount of liquid pool acid and use a tiny bit of that with normal solder. Apply the acid using a toothpick since you only need to wet the surface of the stainless steel.
10 ga wire is enormous. You will be fine using normal 18 ga lamp cord or even 20 ga which is a bit thinner. Solder a clip to one end of each of the wires and twist the other ends together to form two different bundles. Then apply solder to each of the twisted together ends to form a nice attachement point for your launch controller clip. This will create two clip whips.
https://blastzone.org/nar/narts/store.asp?groupid=81620092133327
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