A linear regulator will not work in this application for two reasons.
1. At low speeds, when the generator output voltage is below 14 V, a linear regulator cannot boost it up to 14 V. There are switching regulators called "buck-boost", that can move seamlessly between boosting an undervoltage and bucking (regulating down) an overvoltage. Linear Technology (now a part of Analog Devices) is big in this arena. Know that this is not a trivial design project at low power. At 350 W it is a difficult design project.
2. If the generator can produce 350 W at 24 V, regulating that down means dissipating 146 W. That's a lot.
Please add some details to your question. At what bike speed does the generator output drop below 15 V? When producing 15 V, how much current is available?
ak