How can I deal with a professor who refuses to communicate?

In this sort of impossible situation in life in general, it is generally best not to focus on blame or complaint about someone else's behaviour. Instead (preferably in consultation with a neutral advisor) formulate what you think is the best way forward. In other words, you need to find the solution.

The reason is that no-one (at whatever level) like to be presented with a problem. Everybody likes to be presented with the answer to a problem.

For example:

Dear X

I am having trouble arranging *****. It's difficult to see a way forward although I have done my best. Would it be possible for me to have the following alternative? Blah blah blah. If so I would be most grateful if you could direct me to the right person to help me achieve this. Many thanks.

Speak to your department and explain the situation. Many faculties have directors of studies whose job it is to deal with issues such as this. If you are concerned about irking your examiner(s), you can also mention this to the department and ask for appropriate discretion. They should be able to advise you best and clear this up sooner rather than later, even if it means assigning you a new examiner.

Given the reluctance of anyone else to communicate with Prof A, it seems less a case of you not knowing the secret handshake to get in touch with him, but more a case of Prof A simply not communicating. It is unlikely that anyone (including the dean) can make him communicate in a reasonable way.

What you need to do is to make sure that the path-of-least-resistance for your department is something that ends with you graduating very soon. My guess is that the best bet for that would be to get someone else to examine you. The "Vorsitzende(r) des Pruefungsausschuss" should be able to authorize such a thing. Phrase it as "Prof A is unavailable for examinations", not as "I failed to get in touch with him". If they claim that Prof A is available, ask them to demonstrate by telling you the date for the exam. Be polite but persistent.

If prof. A is indeed as 'odd' to communicate with as you indicate, the problem will be known by the rest of the faculty, probably including the dean. If you want to move forward, you need to escalate. But if you do it in a respectful manner - keeping the focus on your need to move forward, and not on the oddities by prof. A - you can probably manage to do it without hurting any fragile egos too much. A nice trick here, is to blame external circumstances, instead of a persons' inability to communicate.

So: Write to the dean. Tell him or her that you have unsuccessfully tried to set up an exam date with prof. A in these trying times of COVID, but to no avail. Tell the dean that you have already set up an exam date with prof. B, but would like the dean's help or suggestions to set up an appointment with A.

The professor most likely has a secretary taking care of administration. Visit and talk to them, it is likely that they are aware of these difficulties. If you explain your situation and politely ask for help chances are that they will help you communicating with the Prof. to get what you need.