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 shared between many missions, 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 provided by gossamer nets launched from the destination station and into the path of the incoming vehicle. This vaguely resembles an aircraft carrier's on-deck arrest cables ... though the "deck" may be tens of kilometers long.
The station can restore some momentum by mechanically launching other vehicles back down to Earth. However, the station will grow, and construct other space objects, so it must continuously manufacture thrust. If the station is designed to follow a minimum energy trajectory cycle (perturbed by tidal forces over days and months, but returning to predictable positions) then an electrically powered rocket engine can be optimized for relatively low continuous thrust near apogee. It can be heavy, and optimized instead for long term reliability and propellant efficiency rather than low mass and energy efficiency.
An important principle for giga-scale space operations 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 ). A high expansion nozzle can collimate and cool the propellant plume, launching it beyond the magnetopause to be carried away from cislunar space by the solar wind. Assume redundant engines, frequent maintenance, and scheduled replacement of throat liners and propellant pumps.
But how do we hit a very small bullseye (a ten meter net) after a 100 megameter journey in lunar and solar tides? The vehicle will leave the launch loop through a turbulent atmosphere, and a few micrometers per second of unpredictable velocity error will create meters of arrival miss distance.
We can measure calculate the trajectory to a gnat's eyelash - after all, we can launch a space-probe to Mars with superb accuracy, and target a kilometer-high atmospheric window after a 500 million kilometer journey - 9 decimal place precision. We can calculate the positions of the Earth-orbiting LAGEOS Laser Geodynamic Satellites to micrometer precision; indeed, we use those measured positions to define global geodetic latitude and longitude, even estimate centimeter/year continental drift. With such ultra-accurate measurement, hitting a 10 meter net, after an 11 hour, 130 million meter journey, will be a piece of cake.
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 pulse-ablated at very high velocities, with plumes launched perhaps 15 km/s retrograde (relative to the vehicle) near launch time, so they fall back to earth.
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 nets, and laser ablation thrust can help us shape trajectories for vehicles and for the orbital "dandruff" that they shed.