The labor time required to replace the relay is guided by the flat rate manual for your specific model car. The manual lists the R&R time (remove and replace) for the specific part. So if the manual time is 1.5 hours and the labor rate is $50/hr, the charge would be $75.
Go back to the dealer and have the service adviser look up the flat rate time for the part in question and show it to you. They'll tell you the labor rate and you can do the math. They may have used an incorrect time. Most of this is now computerized, they could have made an input error. However, depending on the dificulty of the job and their hourly labor rate, the charge could be appropriate.
The dealership makes money by beating the flat rate time, which are very liberal. So if the mechanic does the 1.5 hour job in 30 minutes, the rest is profit, and that's standard procedure everywhere. I would guess their labor rate is around $75/hr.
There are typically other billed items on the bill such as shop supplies, disposal, etc. However, the labor for the replacement part itself should be clearly reflected on the bill.
You might want them to clarify why the shipping charges were so high. If it's a special order, overnight etc, you might be stuck. But isn't it their responsibility to stock parts? Their overhead is included in the profit they make on the retail charge you end up paying.
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