If the photons of a laser would produce a radiation pressure upon whatever it shown upon wouldn't it be accurate to say that the laser would be propelled in the opposite direction of its beam?
There is a sense in which this is right. If we model the laser as a stream of photons, hitting some surface, then it is the change in the momentum of the photons due to their interaction with the surface that causes the pressure.
For example, if the laser shines on a mirror, then the photons will bounce back after hitting the mirror, and there momenta will change by an amount equal to twice the magnitude of their original momentum.
However, it is also possible for the photons to be absorbed by the material. In this case, the photons don't bounce back, but their momentum change by an amount equal to the initial magnitude of the their momenta, and this momentum change due to interaction with the surface is what causes the pressure.
On the other hand, if you're referring to the object that emits the laser, then it will certainly be the case that this object will feel a force in a direction opposite that of the beam travel direction. By conservation of momentum, if the laser emits a photon of momentum $p$, then its momentum must increase in the opposite direction by that same amount $p$ as well. The change in momentum of the object then leads to propulsion.
See also the wiki and especially the second paragraph in the Quantum Theory Argument section.