It depends on the company. Normally it will not be possible.
You should book via phone and ask which car you can take.
Some smaller Car-Rental companies may let you go.

It wont be easy. You need a WRITTEN PERMIT that you can take your car.
If you take your car w/o permission you DRIVE W/O ANY INSURANCE.
So if you crash the car or they steal it, you have to pay for the car!
It is a risk that i wont consider, never ever.
So why dont you fly to Bulgaria and rent a car there?