a week ago i shipped a car from karachi to islamabad. their contact detail are as follows, call them to ask if they bring car from lahore too.
they have their own car carrier service:
name: subhan goods and car carrier service
person incharge: zafar abbas
contact number: 0300-2678455 , 0321-8781459 , 02132355255 , 021-32355049.
good reliable service, insured also, cost from karachi to islamabad = 22600 rupees including insurance.
hope this will help