House built in allotment garden without building permit (year 1920)

Illegal construction remains illegal construction, no one can guarantee that the TE will receive a decision in their favor.

...therefore I would explore a deal during the conversation. Trial makes wise.

Could one make a deal with the office to allow a summer house, some kind of vacation apartment? And quietly and secretly, the summer doesn't end on October 15th?

What kind of deal is that? Relax in summer and hide in winter? My opinion: Either no risk no fun as an outlaw and thick-skinned through all instances until 'carrying out by the executive or legally binding decision by the authority.

Escroda, did you really understand that? Since the 1950s, the office there has kept quiet about 3 residential units that were obviously built illegally. Now the new owner wants to build, modernize, or change something on one of the three to make it contemporary. That is not possible without approval. Leaving it as is would only be allowed if everything remains unchanged, at most some new paint and such. So if you want to change something, you have to find a solution with the office, explore options. Karsten

However, I have been put off again and again for months there.

My conclusion: Nobody wants to take action here.

Has anyone ever been in a similar situation or knows whether an expansion is legally feasible?

My conclusion: Homer wants legal certainty.

But fine. Let's play "good official, bad official." You are naturally the good one. You are citizen-friendly, understanding, and interested in unbureaucratic solutions. You know that your urban planners are not motivated and that the municipality has no money to draft or amend a development plan. The current planning law does not allow approval as a permanent residence (a preliminary building inquiry might lead to a different result here). So you suggest the summer house idea. The renovation is approved and carried out at considerable cost.
At the local Oktoberfest, while I go to get beer, Homer sits in my seat—away means lost seat! What a disgrace! As a vengeful person and your holiday substitute, I am aware of the deal and, during your winter vacation (or in four years), I issue a usage ban for the summer house in winter in compliance with the building permit. With the necessary cash, overwintering in Mallorca should be comfortable, but the deal is off.

...and what is he supposed to do now?