Subdivide it
The problem you'll run into with any galactic coordinate system is that, for your typical freighter run, you're only going to use a tiny portion of it. Yes, you could use cartesian or polar coordinates based on the location of Sagittarius A*, but nobody wants to describe their flight as a regular run between O 273.334.5616 and O 274.334.5618. You need something that's meaningful locally, but translates to something that's useful globally.
If you're basing your coordinates on a spiral arm, you can start with a linear measure along that arm. You can have your survey crew identify a curve that runs from the beginning of the spiral arm to the end of it, then chop that line into meaningful bits.
The Orion arm is about 3.5kly edge to edge, and roughly 20kly long. The maximum resolution that we care about is roughly 1 ly. You do get stars closer than that, but that's something you can adapt to.
Let's chop that length into 1000 light year chunks and label them A-Z. You could keep the extra letters for later, or give them a special purpose. That's up to you. This gives you roughly twenty zones that are 1000 x 3500 x 1000.
This is where you want to start paying attention to the way political boundaries are drawn up. You could split the boundaries purely mathematically, but you really want to break it up so that things like nebulae and big star clusters get grouped together. If there are edge systems, go ahead and have people argue about which major sector they belong to. It all translates to the same top-level coordinate anyway
So what we do is pick a star in every cluster that is near the middle, and doesn't have a high relative velocity compared to the arm. A large, bright star would also be beneficial. Use that (movement and all) as the distinct, moving center of your zone. You can pick cartesian or radial coordinates from that point.
Personally I'd apply a second layer of letters and numbers, so you're dividing it into 36 x 36 x 36 sub-zones, each roughly 27 x 97 x 27 ly across.
This gets you up to 1.2 million sectors, with labels like Sector Om, sub-sector 2FQ, zone 23.85.21.
Once you have a volume of space, number the stars by magnitude, and planets by distance from their star, moons by (etc. etc.)
The thing this system doesn't do is account for the motion of stars. Fortunately, even today's computers can calculate the current and future location of all of the stars. Ok, maybe not ALL of them, but that's why your people need to trade star charts. You pick a point in time and set the designation, then maybe update it every hundred years. Thus, time stamps would be critical for star charts.