I don't know much about car rental in Serbia these days, I know it used to be very expensive and not cost-effective, but there is a lot more competition now. Why not contact one of the big companies that operate in Serbia (Hertz, Avis) and see what kind of deal they can give you. Leasing might be an option too, through one of the big dealers or banks, though you need some sort of firm status in Serbia for that, like residency or a registered company, which I guess you will.
I agree that public transport gets you a long way in Belgrade, but can understand that for professional purposes you need a car, it's just quicker apart from anything.
I don't know where you are from, but shipping a car from the US, say, used to cost around $1000. There is a fair bit of hassle this end in doing a temporary import and registration, but you can probably get a freight forwarding company to do most of the work for you.
Either that or buy a car locally, but that is going to depend on your budget of course...