How do I find the right wire in this messy ceiling box?

Those large brown and blue bundles are just the main loop-trough for the rest of the house circuit. If you tap the light into that brown bundle, it will stay on 24/7 because those are constant live wires feeding other outlets or lights down the line. In a typical Icelandic setup with that many colors, you are looking at a central junction box where various switch lines are distributed through the ceiling. The wires you need to focus on are those smaller pairs of black, purple, red, or orange. One of those is defintely your switched live. The quickest way to identify it is to pop the cover off the wall switch that is supposed to control this fixture. Look at wich color is connected to the terminal marked with a wavy arrow or an L1 tag. If you see a purple wire there, for example, then the purple pair in your ceiling is your target.

Usually, one wire in that colored pair is the feed going down to the switch and the other is the return coming back up to the light. You will want to connect your lamp's live wire to that return wire. The lamp's neutral wire goes into the big cluster of blue wires, and the earth goes to the green and yellow bundle. It looks like a bird's nest, but as long as you keep the neutral and earth with their respective groups, you are mostly there. If you have a multimeter, you can double-check this in a few seconds. Set it to AC volts, put one lead in the blue neutral bundle and the other on your suspected switched wire. Have someone toggle the switch. If it reads 230V only when the switch is on, you have the right one. Just be careful not to touch the brown bundle while the power is on, since that is always hot regardless of the switch position.

I just pulled the switch plate and I see both a black and a purple wire in there. Does that mean this box might be set up for a dimming system or a double switch?

I only have one of those non-contact voltage tester pens—can I use that to find the return wire without pulling the Wago connectors apart? I'm worried about the pen picking up 'ghost' readings from the constant live brown wires right next to them, since it's so crowded in there.

Ditch the voltage pen for this. In a box packed with live brown loops, those non-contact testers chirp at everythign; they’re pretty much useless for pinpointing a single wire in a cluster like that. You realy need a proper multimeter to be sure. Since you’ve got black and purple at the switch, it’s likely a double switch setup, what we call a krónurofi. One wire probably feeds one part of a fixture and the other handles the rest. If you're just hanging one standard lamp, pick one of those colors to be your switched live and keep the other one safely capped off in its connector.

Sounds like a classic krónurofi setup. Since your new light probly only has one live connection, you just need to pick one of those colors — either the black or the purple — and leave the other one alone in its connector. If you want both buttons on the wall switch to turn the light on, you'd have to bridge them, but that's usually overkill.

Also, watch out when you're shoving everything back in. Those Icelandic ceiling boxes are notorious for being a total nightmare to close. If you force it too hard, you might pop a wire out of a Wago and then you'll be hunting for a loose connection all night.

If you're using those purple or black return wires, just make sure you aren't accidentally pulling on the main brown loop when you're tucking the lamp bracket in. In those shallow Icelandic ceiling boxes, it’s super easy to slightly unseat a wire from an old push-in Wago.

I’d swap any old clear-plastic push-ins for the newer 221 lever-action ones if you have space. They handle the physical vibration of mounting a heavy fixture much better. Also, check if the purple wire is the 'primary' one—sometimes in older homes, one color is bridged at the switch anyway to bypass a defunct dimming module.