On my 94 Cobra with underdrive pullies, minus the alt pulley, it does the same thing. The only way to fix if is to go back to stock pullies. It should be OK, just don't idle a long time at night with lights, ac and radio on, battery might go dead after 15 minutes +. It is something I have learned to live with, I don't have cooling issues so it does not bother me.
2010 Shelby GT500, Kona Blue W silver stripes.
2007 Shelby GT500, Vista Blue with Tungston gray stripes, 1 of 119, Wifes car...
94 Cobra Pace car Convertible # 591
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2001 Cobra vert # 2001 SOLD