I honesty don't know what the best course of action is in this case since I never had this happen to me in my whole career. Any advice?
It just doesn't sound like there's enough money at stake to pay german lawyers, manage the case and come out ahead even if they pay.
You can try a place like this (just a result of a quick google, no idea about them) https://www.bierensgroup.com/en-gb/debt-collection-germany/
Getting chased up in german, from germany, in the correct form might panic them into paying.
If you were actually employed you need to take the company to the labour court (Lohnklage), given how difficult it is to employ someone who's residing outside of the EU I imagine this to be even scarier for the company, so they should be very willing to settle their debt with you.
Again, the key point is to plausibly threaten and possibly enact these measures. This is not legal advice.
Write a short summary of the events, detailing what they should have paid you, when they should pay it, what they did pay you, when they paid it, and how much they still owe you. Also include a copy of your contract.
Use bullet points and make it as easy to read as possible. Imagine you’re asking a stack overflow question.
Probably the lawyer will give you a free initial consultation where they tell you the next free steps.
Maybe the lawyer will charge you a few hundred euros to send a letter before action.
If the company doesn’t pay, and is still solvent then the lawyer might recommend taking them to court as the company will have to pay the lawyers fees if you win the case.
It’s likely your case is open and shut and it’ll never go anywhere near court.