I assume that a 350 miles round trip is going to take all day. So, it has to represent that pay that you want for an entire day's work, plus mileage and at least one meal. 350 miles at $0.58 per mile = $203. Call it $200. Then add an hourly rate for your travel time, including lunch and rest stops. By the time you are done, this sounds like an 8 hour day. What should the hourly rate be? Lets assume $20 per hour, then 8 x $20 per hour = $160. So, $200 + $160 = $360 which is about what you asked for,
When I do business verifications, I price them assuming that it will be an isolated trip, not part of a route. Scheduling is largely at the convenience of the client and is targeted for the next 1 or 2 days. If I happen to find something to add, that is great, but I can't count on it. Plus, with a 350-mile round trip, there won't be much left-over time anyway.
If you do get this assignment, take lots of extra photos. The worst case would be to miss something and then have to go back. I suggest going to the nearest Wifi spot with a laptop and entering the report, but even that does not guarantee that the MSC will not find something they are not happy with days later.
I would really consider renting a car with unlimited mileage. 350 miles is a lot to put on your car and even though the IRS rate is supposed to cover repairs, etc., renting a car for $100 (high estimate) for one day takes the "break-down" risk-off your shoulders.
Last week, an MSC that I do a lot of work for asked me to do a last-minute rush job. It was 5 minutes inside a bank, but 48 miles away. I considered how much of an inducement I needed and gave them a discount for being a valuable client, and did it for $75. Using the rationale that I provided above, it should have been about $100.
Shopping Southeast Pennsylvania, Delaware above the canal, and South Jersey since 2008