I messed a little bit with the jeep and my problems only occur when the interior dome light comes on. I pulled the light and just the under-dash lights don't cause the player to shut-down.

I can't recall if it comes back on shortly after the light is turned on, which would indicate a short voltage drop (negative spike), or a longer one. A longer one would probably indicate that they are pulling from the same source wire, and it could be overloading the current draw from the single source.

I'll meses with addding another power feed from the battery (fused of course). 12 gauge? I've run amps on smaller. I was thinking 16 and maybe 14, if I can find some.