A launch loop can provide very cheap thrust, but can only vary vehicle mass (up to a maximum) and the time (Earth rotation angle) and velocity of vehicle launch. Adjusting trajectories for a later rendezvous will require delta-V provided by other means; electromechanical capture systems or rockets. A fully powered launch loop (6 GWe Earth-surface-provided power) can launch 400 tonnes to high orbit per hour. Given $0.20 / KWh electrical generation, that is an energy cost of $3/kg, or $15,000 for a 5000 kg vehicle.
Even with with the mass production economies of three launch loops launching 2 million vehicles per year at 95% capacity, a thruster package suitable for a Soyuz or Dragon cargo vehicle will add far too much expense. Cargo vehicles should be as dumb as possible, preferably made of plastic and wood, with nothing more sophisticated than two cheap transponders. Guidance, positioning, and thrust should be provided by external orbital systems, such laser interferometers for position and velocity measurement, and high power thrust lasers, which will be highly redundant and re-used hundreds to millions of times per year.
Many (almost all?) vehicles will be aimed at capture systems associated with construction orbit stations; multi-thousand tonne assemblies in geosynchronous (but highly elliptical and not geostationary) orbits. Lunar and solar tides will shift station apogee relative to the apogee of a vehicle launch, and both station and vehicle will need small ongoing thrust to line up and synchronize with each other.
A one day (sidereal, 86164 seconds) construction orbit has a 8378 km perigee radius (above crowded LEO) and a 75950 km apogee radius, a semimajor axis of 42614 km (the same as a geostationary orbit), and an apogee velocity of 1022 m/s . A vehicle launched from an 80 kilometer altitude (6458 km radius) launch loop to the same 75950 km apogee radius will arrive with a velocity deficit of 114 meters per second, which will be added with a catch wire arrest system similar to an aircraft carrier landing system ... though the "deck" will be kilometers long, we can spool up lots of catch wire. We can use the captured energy for power for electric engines, perhaps with some additional velocity supplied by propellant exhaust (hydrogen peroxide?).
Note: one of the goals of launch loop is "no gram left behind". The retrograde rocket exhaust plume should either be faster than escape velocity (3.3 km/s escape at 75950 km + 1 km/s prograde station velocity + Maxwellian thermal expansion plume velocity, ISP > 450 sec ), or slow enough to fall to Earth ( less than 1 km/s prograde + 0.9 km/s retrograde launch minus Maxwellian thermal expansion plume velocity, perhaps an ISP < 180 sec ). These will be fixed rockets on the construction station, reused many times per day. They will have very wide bells for high expansion ratios, and hopefully will have cool, collimated, low thermal velocity exhaust plumes. Assume redundant engines, frequent maintenance, and scheduled replacement of throat liners and propellant pumps.
But how do we hit a very small bullseye (one meter) after a 100 megameter journey in lunar and solar tides? The 5 tonne vehicle may require 20 m/s of delta V (WAG) during the first 2000 seconds at ranges up to 10,000 km after launch, and another 8 m/s of delta V (POMA) over a 100,000 km range during the subsequent 40,000 second journey. Those are 50 Newton and 1 Newton average thrusts. Panels of laser-ablative rubber attached to the sides of the vehicle may be pulsed off at very high velocities, perhaps up to 15 km/s retrograde close to launch (falling back to earth), perhaps as fast at much higher altitudes where the vehicle moves more slowly and Earth escape velocity is lower.
This will be vastly less exhaust plume than a brute force rocket launch, but centuries from now, when mankind launches billions of tonnes into orbit per year, even a few million tonnes of orbiting propellant plume will damage optical surfaces, like the orbiting propulsion lasers and their solar power arrays. If we hope to spread into interstellar space someday, we must plan for interstellar time scales; millions of years across the galaxy, at speeds we can afford to slow down from at the destination. We must not imprison the Earth in debris rings (particulate or molecular) in a tiny fraction of that time. Launch Loop, capture wires, and laser ablation thrust can help us shape trajectories for vehicles and for the orbital "dandruff" that they shed.
SESAME materials database LLNL