First, please go to an inspection station that does not have a vested interest in quoting you for expensive repairs. Search google reviews for the most lenient inspectors. Try a low-income area. Low income people drive cars like this all the time, somehow they pass inspection. Do that battery yourself at an auto parts place.

You stated that the lack of inspection is the real issue, otherwise I think you all would be driving this beater and you should try to drive it into the ground.

You may also try a private party sale to see if you can get enough to pay off that loan. Disclose to the buyer it won't pass inspection, you might find some one who will repair on their own.