What to do if I had my engine rebuild by a transmission chain but a few days later it caught on fire?

It would be advisable to get a statement in writing from the mechanic who said the engine was wrongly rewired.  Show that written statement to the transmission chain.  If the transmission chain is not willing to make the necessary repairs to replace the engine and compensate you for the damage to your car, sue them for negligence.  Negligence is the failure to exercise due care (that degree of care that a reasonable mechanic would have exercised under the same or similar circumstances to prevent foreseeable harm).  Your damages (the amount of compensation you are seeking in your lawsuit) would be the cost of repairs to your car for the loss of the engine and damage to your car from the fire.

You will need to mitigate (minimize) damages by selecting another repair shop whose charges are comparable to other repair shops in the area.  If you were to select the most expensive auto repair shop you could find, your damages would be reduced accordingly because you failed to mitigate damages.  If you need a rental car while your car is being repaired, your damages should also include the cost of the rental car.  Again, you will need to mitigate damages by selecting a rental car with a reasonable rate.  If you were to select the most expensive rental car you could find, your damages would be reduced accordingly because you failed to mitigate damages.