In streaming mode, the maximum possible sampling rate for the PicoScope 2206 is around 10MS/s shared between both channels, so it probably won't be suitable for your application.
In addition, the maximum streaming rate is dependent on your PC's available resources, so with a 3rd party application such as NI LabVIEW running, you may not achieve this maximum rate.
In streaming mode, only a single trigger is possible - it is not possible to re-arm the trigger until data collection has stopped. This is typically for use in situations where you are continuously monitoring a system and want to stop data collection on a particular event.
For your application, the PicoScope 3400 Series range of oscilloscopes might be more suitable (1GS/s with a single channel, 500MS/s with two channels enabled):http://www.picotech.com/high-performance-4-channel-oscilloscopes.html
These oscilloscopes also have a relatively larger amount of buffer memory. For example, the PicoScope 3404A has a 4MS buffer shared between all channels. This would be enough for 10 waveforms of 120 samples each.
I hope this helps,