Unless you're going to squeeze 10 seats or more in there or operate the airplane in a commuter (scheduled) operation, you need only a second class medical certificate and a commercial pilot certificate.
Why are people replying with "I believe" or guesses, when the regulation is so clear, and so basic?
14 CFR 135.243(a)(1) identifies the specific circumstances under which an ATP is required for operations under Part 135. A first class medical certificate is only required to exercise the privileges of the ATP. If the ATP is not required (in this case, it isn't), then only second class privileges are necessary. This may be a first class medical certificate that is older than six months (after the month in which the exam was taken), or a second class certificate within 12 months after the month of the date of the examination.
I'm all for asking questions, but read the regulation before asking, and when replying, refer to the regulation and read it. Don't just guess. That's not consistant with our profession.