As someone who always rents a vehicle, the cost of fuel is nominal. The vehicles over there get very good mileage. For a 12 day trip you’d probably only have to fill up once unless you’re driving all over the countryside.
Whether you’d need a vehicle would depend on where you intend to stop while chasing the eclipse. If only visiting cities serviced by train or bus, then you wouldn’t need a car. There are a lot of villages in the Burgos area not serviced by train worth visiting and a vehicle would come in handy to visit them. My guess is you will have some non-eclipse time where you could visit places.
You are correct that parking in cities is tight and can be expensive. If your lodging has parking available it will be less than a parking garage. The last time I stayed in Burgos parking at the hotel was 15€ a night if I remember correctly. However, the last time I visited there in 2022 I parked in a free lot about a 10 minutes walk from the cathedral.
There are free parking lots in most cities, including Burgos, but you need to know where they are as they are generally not marked as being free. If they are marked you need to know some Spanish to know they are free. I look them up before a trip and bookmark them to Google maps. Parking in small villages is mostly free although you may need to walk a few minutes to get to the historic center. Again, I research parking lots before trips so I can go right to them.
I wouldn’t wait too long to make reservations as a lot of people will be doing the same as you and lodging prices along the main eclipse axis will most likely be elevated. Some hotels may already be fully booked. Don’t overlook lodging outside the large cities.
