[Traylaw]

You need to get the error code to see why you have an SRS warning. Your dog riding shotgun would not cause this but a component failure does. You can go to most auto stores and have them pull the code at no charge.
If you are referring to the SRS indicator in the center of the dash next to the IMD, this is simply an SRS enabled/disabled indicator based on the weight seen in the passenger seat. Nothing to reset here, the sensor is working, not working.

 

[Traylaw]

If you are looking for a OBD scanner that can read and reset SRS codes, Bluedriver a good choice that works with your smart phone or tablet. They sell for $99.

 

[ms3224a]

The problem needs to be fixed. Why? It's a requirement to drive with the dog in the front seat. How else can they stick their head out the window while you're driving? Since she's been banished to the trunk, she probably thinks she's done something wrong.