Negotiating Part-Time Lecturer in Germany
Given that the position as you describe in the comments "offers the possibility to enroll in [their] PhD program or to complete a habilitation", this should be interpreted as a typical qualification position in the German academic system. The concept of these positions is that you are an employee of the university, with some tasks in teaching, administration, and/or research, but you get time to work on your next qualification in an academic career. There is a dedicated federal law (informal name "Wissenschaftszeitvertragsgesetz", officially "Gesetz über befristete Arbeitsverträge in der Wissenschaft") that governs these positions.
In some fields (yes, usually the less well funded ones), it is quite common that only the "tasks" part is funded, but you have to do the "qualification" part unpaid, especially when working towards a PhD. Nowadays there is indeed a recommendation to fund all such positions for least at 65%, but given budget constraints etc. you can not always count on that. Nevertheless, with this type of position, the university needs to make sure that you have sufficient time to work on your qualification, so in principle the setup should be such that you need not more than the ~ 20 h / week to do the administrative / teaching part. Depending on the culture of the research team you're joining, you may still be expected to spend most or all of the "qualification time" there, so things may get a bit hard to keep apart.
The HR department will not negotiate with you about the part-time ratio nor about the salary group (such as E13 or E14) for this position. That is decided by the person that is filling the position, taking their budgetary constraints into account.
What the HR department will indeed do is determine the salary level (step 1, 2, 3, ...) that you are rated at. However, step 2 typically already requires at least some years of experience in a very similar position, and step 3 or 4 is indeed next to impossible, as you already figured out.
My recommendation would be to only take this type of position if you are more interested in the qualification part than in the teaching / administrative part. In the specific case you describe here it worries me a bit that the qualification part seems to be hardly described in the posting, which may indicate that the university "abuses" a qualification position to get some core teaching / administrative tasks done cheaply. At the interview, definitely ask about the research part, i.e., what topics you could work on, or come with some ideas of your own, and if they don't seem interested in that it's a big warning sign.
If this is for doing a PhD and you're in principle qualified to do that in Germany, then it should not be too difficult to find a position where the funded part is a small teaching support role or even working on a funded research project that can be connected to your PhD topic. Any of these would be much more preferable for doing a PhD than the type of position you describe here.
From your description I suspect that the university received extra funding for just half a (probably E13) position. That means HR cannot increase that; the money just isn't there. Apparently one of the profs is willing to look for additional funding, which is your best bet for increasing the hours. However, additional funding also means additional tasks. With HR you can probably only negotiate minor steps in your pay scale. That is important, but is not your question.