If you can spark it with the lasers, can you keep it going by feeding it with a collider?
No. People call it "burning," but it isn't like lighting a log in the fireplace.
Even if you could feed a collider continuously, target can't be. Inertially confined fusion depends on a fuel pellet getting compressed by a uniform implosion. That implosion is fed by lasers conventionally, and we did it with heavy ion colliders. Because it starts with a pellet, it's inherently pulsed.
Magnetic confined fusion (like Tokamaks) offered the promise of a continuous burn, but that has yet to materialize.
Obviously, it works fine in the Sun, using gravitational confinement.
You can feed a collider continuously IF you can keep a constant electrical field there to accelerate it. That has power implications, so these things are usually either low energy or highly pulsed. You also get one pass for a DC field, so it has to be a BIG one. If you try to accelerate past a DC field multiple times, you lose all the previous accelerations on the way around. What's typically done is that the electric field is oscillated at the same frequency the beams circulate, which makes them "bunchy." That way, you can get a kick as you go through the electrodes, get another kick on the way around the ring, then, get the same kick going through the electrodes again. But that's not continuous. Particles that are out of phase with the RF will get decelerated and lost rather quickly. That is, however, the essence of the original cyclotron design by Lawrence, and it is still heavily used. ALS uses a much more sophisticated array of oscillating electric and magnetic fields, but it's essentially the same idea. It gets a lot nastier when the particles get Relativistic, as the time dilation effect changes the RF frequency. You can see the compensation for that if you ever take a close look at the SLAC collider -- the acceleration chambers are longer further from the source. That makes it pulsed as well.