What a great idea to combine a cycling trip with driving around Costa Brava. This is a great way to experience the region’s cycling landscape, history, and coastline. It is indeed possible. My favourite Girona itinerary for biking and driving starts in Ripoli, about 40 minutes northwest of Olot.
Bike section - start in Ripoli
Start your cycling section from Ripoli, and bike southeast to Girona via Joanetes, over a 2 day period. Spend your first night in Joanetes, and your second night in Girona. There are many bike friendly hotels in Girona.
Visit Girona for a day and get a rest from cycling. I like this Girona 1 day itinerary.
From Girona, bike east towards the coast to Sant Feliu. Spend the night in Sant Feliu.
Next morning cycle up north hugging the coast to Begur. Be prepared to cycle along with some traffic because this part of the region is not really cycling tracks. The good thing is that you can break up the trip into sections and ride via some inland villages. In particular, Palamos to Palafrugell has a good cycling route but San Feliu to Palamos is mostly by road.
Hire a car in Begur, don’t rely on the big global companies and instead choose local ones who will be flexible with your needs. From Begur drive around to Figueres, Cadaques, and Portlligat. Perhaps add Pubol to make the Dali triangle complete. You can spend the night in Cadaques, or any town that you like in this area.
On another day visit Besalu and Santa Pau. Depending on how much time you have, take another day to visit Sant Marti d'Empuries and Roses. Return to Girona or Barcelona.
Drop off your cycles and equipment in Begur if possible or else you will have to drag them around for the rest of the trip (which is also fine if you want to explore the countryside cycling routes as you drive around).
You have a fair concern for equipment safety - not that the region is unsafe but you obviously want a smooth trip.
Don’t worry because this is a very often asked question to all hotels in Girona. My suggestion is that you ask your hotel if you have already booked a place or have some options in mind. Choose the hotel you like and then ask them, it is simpler to do it this way.
My second suggestion is to book an apartment with a balcony - it is quite safe in Girona especially in your own apartment - hosts are used to visitors who bring their cycles and equipment. Athenou is a good option.