OK done some calculations for the cyclotron approach and it's quite scary!
The centripetal force is given by the following equation:
F = m * v^2 / r
So lets say your cyclotron is the size of geostationary orbit around Earth. I'm not saying that's where you would put it but the bigger it is the smaller the centripetal force so lets make it big! So the radius of geostationary orbit is around 40,000 km or 40,000,000 m. Lets look at the force when you reach .5 of lightspeed, that is around 150,000,000 m/s. Finally let's say your mass is around 1000kg; optimistic for a starship but it gives a starting point.
This gives a centripetal force of 562,500,000,000 Newtons (I think I've got all the units right?). Could such a centripetal force be controlled in a cyclotron? And how?
Bear in mind your ship mass would probably be greater than 1000kg, your cyclotron would probably for practical reasons have to be a lot smaller and we are really looking for 70-80% of lightspeed for your final velocity. All of which would only increase the centripetal force to be managed.
I am becoming less optimistic about this approach