I would question whether the 300W figure mentioned is correct; this is a significant amount of power to burn off in a single transistor, and in practice it’s not very practical for devices in packages less beefy than an SOT-227.
A correct understanding of the actual amount of thermal power to be dealt with is the first step in identifying a suitable thermal solution. There’s no point in proceeding further until this information is understood clearly. A transistor that’s transmitting 300W of power electrically is a very different thing than one that’s dissipating 300W thermally.
I’d state with confidence however that peltier devices are not appropriate for anything even remotely approaching the amount of power mentioned. Heat pipes probably wouldn’t be a great solution either. If one’s actually needing to deal with 300W*16=4.8kW of thermal power, that’d be in the territory of cold plates and fan-equipped radiators of the sort found here.
A discussion on making thermal analysis calculations that might be informative can be found here.