In my - limited - experience... the smaller the cutter, the faster it should be turned. A 3/8" cutter is small enough that you should be able to dial up the speed to 2000+rpm. With the higher cutter speed comes better rigidity of the tool, so you can cut deeper & faster - you should be able to cut full depth in one pass at that speed; but even if you wanted to go carefully & take say 0.050" passes, it would be fine.
A ball-end mill is a funny beast, because it barely cuts at the bottom, it sort of "smears" the metal; With your 0.005" cuts, you'd be trying to use the least able surface of the ball mill to do the bulk of the cutting. IMHO, you'd be better off using a thinner (say 1/4") roughing mill & go "almost" to depth (say 0.150"), then use the ball end mill to finish off. As Kenneth suggests, if you can use a smaller ball mill (i.e. cutting on one side only), then you'll get a better finish at the expense of taking a bit longer over it.