From personal experience I think you'll find it difficult to rent a plane for 3 weeks straight, most rentals double up as trainers, when I rent in the US 1 day maybe 2 is the longest I can take a plane off the flight school line for.
Where I rent from to take the aircraft for the day means paying for a minimum of 5 hrs regardless of how many hours you fly in that day. Example I've flown 2 hours down to Key West stayed overnight and flown back 2hrs the next day. but paid for 10 hrs.