Cars you could get in your budget.
Alto (rarely available on rent in Lahore)
Cultus (lots will be available, the EFI will be expensive than VXR)
Santro (again, rare on rent, if you get on please check it thoroughly, specially starting & missing, if any).
Do not get Mehran in any case, even if it's 2010.
Coure (would be best choice, very economical and under your budget, only compromises passenger limit)
Vitz (check starting & missing if any, specially CNG Automatic one's)
Things you should check before the office give you the keys.
check oil condition (should be full & thick)
for EFI car check air filter, and check ignition response, plus check patrol as EFI don't start on CNG.
check body (start from bumper, fenders, doors, bonnet, roof & look at the side rails under the doors)
check the tires condition (do not hire the car with worn out tires)
check the jack & bolt tighter, (should be functioning & bolt tighter must be in good shape to grip the studs)
get the rental in your name, this will help you at all check posts, even if you don't have car's original registration book.
If you could wait for 10 minutes at rental office, ask them to wash the car, (they will do it for free, but you will have to get it washed when you will return it).
check all lights & indicators if they work, check the wipers, check if engine warning light does not go away after a start, check brakes & brake fluid level (if any single thing is not working ask them to repair it, they will do it for free, go with them to check if they'r resolving the faults properly or just for time being, proper work will save you from any fault afterwards).
Ask them if the car has a tracking device in it, and ask for the mobile phone number of the person who has the password of the tracking device (in case you want to go somewhere outside Lahore).
After getting the car, go to nearest police station, & tell them your purpose of visit & ask them if they want to issue you any letter of travel with the car's registration & rental office address (just in case you don't have any relative or friend in Lahore).
Driving in Lahore is much civilized, at it is in Karachi or Islamabad. Just cooparate on check posts. In night, put lights on park, & turn on the cabin light for them to look at you from distance. If they ask you that your ID card is of Karachi, tell them you have home here as well (friend or relative).
& best of luck for the trip & holiday.
regards.