Fiat Punto, workshop service and too small battery

Hello, I have a little problem and since I don’t have much idea of cars myself (typical woman just ), I hope that someone can answer my question here. Actually, the problem is more or less solved, I would only like to have opinions about it. And this is about the following: I bought a used, 11-year-old Fiat Punto from a workshop in September 2014. Had only 58 000km and many was new. Now I had problems already in December with the battery: there were problems at the start and I had to start my car once, so in the second gear roll down the mountain and let the clutch come. When I then went to the workshop it was said that it was nothing, only the battery would weaken a little bit, but not enough to do something. Now I wanted to drive to work last week, then my car suddenly didn’t jump at all (I drove with it the day before and it had 20 °C, so that was it I didn’t want to tow my workshop on 50km and they didn’t give me a remote diagnosis either. A friend then said, after I tried out a few things, that’s definitely the battery and gave me starter help. I then drove with all warning lights and without brake amplifiers, etc. to the workshop. They then changed the battery as trigger. However, the new battery is a little bit small and fixed with the help of a wooden block, so that it does not slip, because probably “Fiat batteries have a certain size, but it can also be done with this construction”. I was calculated now almost 70€ for the battery and 30€ service although I still have guarantee. Now to my question: Is this really okay with the battery and would I come with it also by the TÜV? And are the costs justified, where I nevertheless exchange up to the battery itself or my own. Friend had to do everything? Greetings and thanks for your answers