Lost original keys, if I use a transponder chip key, will the car start?

Without being programmed into the system no key will work. A mobile locksmith should be able to make and program in new keys for you as the dealers seem to be pushing you away. That car uses a PATS system that has the security module inside the PCM. So even changing out the locks wouldn't make a difference, you would still need to get the keys programmed into the system to get it to start. If you get two keys programmed into the system so they work, you can use them to add others. Have the smith erase all the old keys prior to adding the new ones, that way if someone else finds the old keys they still won't work to start the car.