For some reason every time I close the readius and get back to gameplay, my character is naked for like a split-second. I know this is very minor; however, it is really annoying. Any explanation/fix for this?

Have you any sort of similar issue on the the mod page, or its comments?

If not you might want to try the following:

-Re-download/Re-install the mod
-Verify the game's cache integrity through steam
-Try another pipboy replacer and see if you encounter the same issue. http://www.nexusmods.com/fallout3/mods/16836/?
-Run the mod through F03 Edit to see if there could be a problem as well.