From a more technical description, when the dome switch is on DOOR, the body ECU has control over the ground feeding the dome light. When the switch is set to off, the ground is totally disconnected from the light. When set to ON, ground is fed straight from the body electrical ground, making it consistently on.
So, if the dome is not set to DOOR, nothing can control it, not even the door, so SKS certainly can't.
Also, be sure SKS is turned on.
And, as stated, the door has to be locked in order for the Prius to start polling for the fob unsolicited. Then, and only then, the dome light will illuminate when you approach the frond doors.