I think the reason the insurance company won't release the car to you is because they declared it totaled and will not insure a totaled car.
If you really want to keep the car, you can ask how much to buy it back from the insurance company. But then again, which insurance company will insure the car without it undergoing repairs?
When my car was totaled, I pretty much signed over the title and received a check that afternoon.
I'm going to guess that the accident was your fault since it's your insurance company that would pay for a rental? That sucks. A lot of times, rental companies have weekend specials or week long specials. I'd recommend searching deal type forums for that info (and coupon codes).