I don't know if the dealer can even do that. The security aspects of the system are extremely overdesigned (IMHO) so that you can end up with anomalous failures (according to dealer tech I talked to when I got mine fixed the first time). As I stated before (ebay listing) you need to transplant the following all together:

ECU (must match car model and engine)
EIS
Keys
ESL

The dealer can sell you a new EIS and code it but you may need new keys also based on the age. For the record I never heard of the symptom you state, only the usual "car won't start". I was unaware the EIS even switched power in the way you are describing. However I have one here and it does have 4 large "Faston" type connectors which at first I assumed were standard 1/4" but are actually about 7mm wide. You could try removing it and taking it apart and cleaning whatever you find inside that looks like a contact (if any). It is a bit of work to remove it. You need a special tool to unscrew the trim ring, then there are about 4 different connectors on it and a fiber optic light connection. That would be the first thing I would try since everything else is a lot more work and money.

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1998 C230 330,000 miles (currently dead of second failed EIS, yours will fail too, turning you into the dealer's personal human cash machine)
1988 F150 144,000 miles (leaks all the colors of the rainbow)
Previous stars: 1981 Brava 210,000 miles, 1978 128 150,000 miles, 1977 B200 Van 175,000 miles, 1972 Vega (great, if rusty, car), 1972 Celica, 1986.5 Supra