I found an academic article (Vogel et al 2019) that analyses this phrase. Key points:

• when the German legislator uses geschäftsmäßig, this demonstrates a clear difference in intention from gewerbsmäßig or gewohnheitsmäßig

• the article quotes Franz von Liszt 1881, and this definition seems to be accepted to this day:

  Die Gewerbsmäßigkeit charakteriſiert ſich einerſeits durch die auf öftere Wiederholung gerichtete Abſicht, andrerſeits durch die Abſicht des Thäters, ſich durch dieſe Wiederholung eine, wenn auch nicht regelmäßig oder dauernd fließende Einnahmsquelle zu verſchaffen […].

  Die Geſchäftsmäßigkeit teilt mit der Gewerbsmäßigkeit die auf regelmäßige Wiederholung gerichtete Abſicht, dagegen fehlt die Abſicht, ſich eine ſtändige Einnahmsquelle zu eröffnen. Ob die einzelnen Handlungen honoriert werden oder nicht, iſt gleichgültig.

• the term geschäftsmäßig is significant for §5 TMG, but has also reached wider attention in the discussion around the decriminalization of assisted suicide.

So the key defining aspect is the auf regelmäßige Wiederholung gerichtete Absicht, the intention directed towards regular repetition.

This meaning in legalese German is divorced from everyday language.

§ 5 TMG has the interesting construction of “geschäftsmäßige, in der Regel gegen Entgelt angebotene Telemedien”. So the TMG does not seem to care whether you have a profit motive, only that other people might provide this kind of service for a profit motive. If other people would provide instances of Discord bots in order to get donations, that might already bring you in scope.

This is not legal advice, but it seems like your options are to either avoid running an instance of the bot, only running it in a private context without access from a wider public, or sucking it up and providing the necessary documentation.

And no, it is probably not possible to use a PO box because you don’t live or work at that address. The general expectation seems to be for the address in an imprint to be ladungsfähig, so that you can be served there. This random lawyer’s website writes:

Unter der Anschrift in diesem Zusammenhang ist die Postleitzahl, der Ort, die Straße und die Hausnummer zu verstehen, nicht ausreichend ist die Angabe eines Postfachs.

That’s not the correct criterion. There are multiple German laws that require imprint-style disclosures.

Some of them are indeed specific to commercial activities.

But the Impressumspflicht typically means §5 TMG which requires an Impressum for

geschäftsmäßige, in der Regel gegen Entgelt angebotene Telemedien

Rough English translation:

Telemedia offered in a business-like manner, typically for remuneration

Critically, “geschäftsmäßig” does not mean “commercial” or “profit-oriented”. In particular, nonprofit organizations also act geschäftsmäßig.

IANAL, but it doesn’t sound like your service wouldn’t be geschäftsmäßig.

All of this is irrelevant anyway because you very likely have to publish a privacy notice per Art 13 or Art 14 GDPR. This must include the identity and contact details of the data controller (i.e., you). The German data protection authorities expect that the identity includes your real name and a ladungsfähige Anschrift (address where you can be served), so pretty much exactly what would be included in an Impressum anyway.