Since this question will probably involve some laws, I will state that I am German and I was booking via an online site which is mediating for travel agencies.
I booked a hotel on May 15 for a one week stay in Italy at Lake Garda (June 19 to 26). On May 25th I was told by the travel agency that the hotel was overbooked and offered me a new, in my opinion inadequate, one to stay in. I told them that I know that region and that the other hotel is not in question for us. I supplied them with a few alternatives, one of them being in the same town, the others in different towns (probably 15 km away).
Today, on May 29, I got a new offer from the agency for one of the hotels that I proposed, but to a higher price. It does have an additional star compared to the originally booked hotel, but it is not in the town I wanted and the town does not even have a beach. They offered me to pay 773 € for the alternative that I suggested when I explicitly said that my budget will not exceed the 653 € I initially paid.
They also offered me a hotel for no additional cost. It also has an additional star and does cost more, but it is on the complete other side of the lake! I specifically looked for a hotel on the east side and driving from Limone takes at least an hour to get to where I want to go on my holiday.
I do want to get a solution from the agency, since it is way too late now to find anything else.
What are my rights regarding compensation if I do accept any of their offers, even if I take the more expensive hotel at no additional cost? Can I somehow demand they let me stay in the (alternative and more expensive) hotel I proposed for no additional cost?