None of the supported modulations is pure PSK; the closest you get is MSK, which, by mathematical happenstance, is like O-QPSK with a halfwave pulse shaping filter. But due to the offset nature, that's not really PSK at all, in general.
we think our signal is PSK modulated, 305 bits per second
I'm taking a wild guess from your signal plot that this might be QPSK. That would give us a symbol rate of 152.5 bd. That's nearly incredibly slow. Are you sure? This isn't anywhere fast enough for speech transmission, so this is only about some control messages. Are you perhaps working with an FM handset (very likely, there might be regulatory rules that mandate it does FM) with something inside? Your signal plot is really not that enlightening, as it looks like it hides the information as the "small" oscillations modulated atop of the "large" oscillations by being zoomed out too far.
Under the condition that we're actually dealing with a receiver that has applies a filter of little more than 150 Hz bandwidth, the offset component duration in the yard-stick-synthesizable MSK might actually, with a healthy degree of luck that the pulse shape is OK for your receiver, be negligible, if you "just" used a large multiple of your needed symbol rate as the symbol rate of the yard stick, and repeated each symbol as often as that multiple dictates, so that there's no change of constellation point (or, conversely, just continuous wave in MSK) for most of the 1/152.5 s duration of a symbol, and just a tiny fraction between these low-rate symbols that looks a bit strange in how it changes from one phase to another.
Generally: wrong tool for the right job, I'd say. Maybe look into proper SDR frontends.