Now that you've looked at a map, it looks like you've also just seen a calendar.
A single week was not possible for the trip to the islands that you want to do.
Now, about the car rental.
As you have noticed it will be very difficult if not impossible to find a rental agency allowing you to make your trip with a single car.
And you can thank them, they are doing you a favor by denying you this possibility.
Here's why:
1/ Few car rental agencies accept that you take the car on a ferry. This will limit you considerably in the choice of rental companies.
2/ Those who accept will make you pay a high price
3/ If there is the slightest problem with the car, you are in a hell of a mess to get assistance
4/ With your travel style you will not drive it but see it parked most of the time in the garage deck of a ferry or travel the 3 kilometers between the port and your accommodation.
5/ In the ferries you will pay the ticket for the car
6/ If we add up the ferry hours, you will pay the equivalent of several days of rental while you and your car are on a ferry
7/ Not all ferries take cars, this will severely limit your travel options
If you are traveling in the coming days, contact local rental companies in the places you are going to visit. Give them your day and time of arrival, your day and time of departure.
Many have the possibility to book online, with other rental companies you can book by email.
The prices for a small car range from 25€ to 45€ / Day depending on the islands and the rental companies.
personally I always choose rental companies that accept cash.
For example this month I rented cars several times on the islands. I didn't pay anything in advance, everything was booked by email. Taking the car lasted 5 minutes, (the last rental company did not even need to see my driver's license)
The return of the car takes only 2 minutes. You give the keys, they briefly look at the condition of the car and that's it.
For 2 weeks rental of category B cars I paid a total of 272.80 Euros (in cash)
It was for the car part.
For the choice of the islands there would be a lot to say too.