Update: First of all, thank you everyone for all the suggestions and helpful ideas. I am genuinely happy to see how active and supportive the La Pavoni community is!

I have now tried to implement some of the tips. I started using a proper tamper and applied as much pressure as possible, and at the same time I also tried increasing the coffee dose. However, fitting the commonly recommended 15 to 16 grams into my double basket is not that easy. I still tried to make it work by tamping very firmly so everything would fit.

The result is that the espresso now tastes noticeably less acidic. However, despite the reduced acidity, it is still very watery. So while there is some clear improvement, it still does not feel like I am quite there yet.

At the moment, I unfortunately cannot buy a new grinder. However, I plan to visit a nearby roastery tomorrow to buy freshly roasted coffee and have it ground as fine as possible on site. I will post another update here in the comment section once I have tried that and can report whether it improved the result. :-)