It depends entirely how the cabinet is rated. I do not know the specific details aboutthe cabinet you have, but if the 300 watt rating is the "RMS" rating (more or less the thermal capacity of the driver along with the average mechanical capacity) you should be just fine.
Now for a little more in depth comments (indulge my tech-speak just a little bit...

)
A speaker's power handling capability depends on both thermal and mechanical limits of the driver(s). The thermal capacity is pretty easy to measure and extrapolate, and say a steady state 300 watt thermal capacity would be very comfortable with a 600 watt or higher amp that was NOT driven into clipping or limiting. This is because the bass guitar signal is not steady state and the peak to average ratio of the bass guitar signal is such that the peaks are typically around 6dB (at least) greater than the average. So, (in theory) a 1200 watt RMS rated amp driven by a (typical) bass guitar will result in a 300 watt (Pmax/2) average thermal load to the speaker.
Now, here's where the "in theory" part does not translate well into practice. It's a pretty optimistic assumption that the speaker's mechanical structure will handle 2X or 4X the thermal power rating. It used to be that the 2X rule of thumb was reasonable, but the power levels were much lower and the speaker ratings much more conservative than today. Also, the mechanical rating varies a LOT with frequency, so if frequencies below the cabinet's tuning are allowed to pass, you enter into a reigon of the driver's mechanical rating that (may) fall off pretty darn quick. In fact, a 300 watt RMS rated driver may have only a 150 watt or lower mechanical rating at say 25Hz. Exceeding the mechanical limits of the driver just causes prematutre failure of the suspension components.
Adding a compresor or limiter seems to be the "easy" fix. Well it certainly helps with mechanical issues but also causes a shift in the peak to average ratio of your signal (that's exactly what these dynamics control devices do) and now you have to reconsider the thermal limitations again.
Regarding the "peak" ratings of speakers, they are IMO a bunk spec, misleading in that the peak power value of a signal is exactly double that of an equiv. "RMS" value. It's purely a mathe expression, just another way of saying exactly the same thing. 300 watts "RMS" is exactly 600 watts "peak".
Now for the good news... your Shuttle amp has two features that help preserve your speakers under high power operation. There is a steep high pass filter (used almost exclusively in the pro audio world) that limits the potential for mechanical speaker damage, as well as a soft clip/limiter that emulates how a tube amp overdrives. With a speaker rated at 300 watts "RMS" I would not expect any problems, even driven into moderate limiting.
Hope this helps.