Assuming that the player doesn't have any problems with its main constant power lead (blown fuse etc), and assuming that it's a Mark2a player instead of a Mark2, then I think you can do what you're asking. I think the bulk of the current is drawn fron the constant power lead, and the ignition-sense is only a sensor line and draws very little current.

However, if your player is a Mark 2, there is a chance that you could draw too much current. There is a difference in the power scheme between the 2 and the 2a described here. If you blow the fuse on the constant-power wire of a Mark 2, then it will draw its current from the ignition switched wire.