Imagine a platform in space. On one end is a sheet vertical. Fixed to the plate is a laser. The laser shines on sthe heet. Newton's 3rd Law wouldseem to say there will be an equal and opposite reaction through the mounts to the plate. Like standing on the plate and pushing against the sheet. Make a fist and start punching the sheet and the net change in momentum is zero. Or throw baseballs at the sheet.
The sail you push on isn't fixed to the laser. The sail isn't fixed to anything but the payload. So it accelerates away from the laser.
Yes, that puts a reactive force on the laser, which will accelerate away from the sail with equal and opposite momentum change, unless there's some other force on the laser. So you'd engineer the thing to have another force on the laser. The obvious arrangement would be to add a solar sail fixed to the laser. So the sun pushes on the solar sail (and also powers the laser), the solar sail pushes on the laser, the laser pushes on the starship's sail, and the laser beam pushes the laser back away from the starship toward the solar sail, balancing the forces on the laser, so the laser stays still. The net overall effect is that the starship accelerates away from the home star and a bazillion photons bounce off the two sails, back in the general direction of the home star. Their collective momentum away from the destination star is what balances the starship's momentum toward the destination star.
The conversion efficiency of energy from sunlight to laser is probably so bad, that there are better ways to keep the local laser in place than a light sail. If the laser platform is close to the sun, you can resupply it with a propellant from time to time. I suppose there could be a chain of lasers along the way, using e.g. nuclear fusion for power, but even then, it's probably more efficient to use
that energy source to shoot ions or something else in the opposite direction to counteract the force from the laser, than it would be to have another laser somewhere in the inner solar system.
Besides, assuming that the laser is powered by solar panels, having some of that real estate be taken by a solar sail that just does nothing except block out the sun seems very impractical.