A dealership damaged my car. Any advice?

There's a lot of information I could share about this. But I do want to remain semi-private, so I'll try to stick to the highlights and answer questions as needed. Instead of stating hard numbers, I'll compare against the value of my car according to Kelly Blue Book.

I'm reaching out here because I'm not a "car person" and I haven't gone through anything like this before. I figured if there's any other angles or strategies I'm missing then you all will probably fill in my blind spots. I want this to be over ASAP, but I also don't want to pay for the damage they've caused since it's thousands of dollars.

What happened:

• In May my car started having acceleration issues. I brought it in for repair at a dealership I've gone to for 2 years who has an in-house service department. The dealership's name includes the name of my car's manufacturer, so they're a specialized licensed dealer.
• At first they thought my catalytic converter was the issue, so they ran a bunch of tests. After extensive troubleshooting, they ended up taking it off and confirmed there was no damage or clogs. I received written updates, so this is well-documented.
• There is a known defect with engines in cars like mine. Therefore they put in a request for a free engine replacement, which was approved. They also reexamined my car after the approval was granted and recommended replacing another part at the same time. I approved and paid for that myself.
• When they gave me back my car, it was making a lot of weird sounds. I commented on it during pickup, and they told me not to drive too fast. Turns out it wasn't able to go on the highway at all and the check engine light quickly came on. I notified them and they asked me to return it. I put 100 miles on it during that period. For reference, driving on a highway has never been an issue for me.
• Upon examination, they informed me my catalytic converter was fully clogged and that it needed to be replaced. The sent me a quote for half the value of my car, a major expense.
• At first they tried arguing that it passed their test drive. But after I pointed out that I only drove 100 miles, they said it must have been clogged the whole time and they missed it repeatedly. They also said the new engine could in theory cause the issue, but stopped short of recommending it be replaced yet again. They also told me I could take my car somewhere else to get the repair done for cheaper. I have this conversation recorded.
• So I haven't had a working car for 100+ days now. It's still at the dealership and they've been trying to pressure me into taking it away, even though it's not save to drive and driving it could cause more damage.
• I've asked for a loaner vehicle repeatedly, and I'm always told none are available and that I'm on the wait list.

Here's what I've done so far.

• I have a support case with the manufacturer's national customer support system. I don't have my case managers direct line, but I call the regular line every day asking for updates. I also ask for it to be escalated and expedited. Sometimes the person tries to go above and beyond to give me the latest updates, but ultimately I don't think this is helping much. A request for a "good will" repair has been submitted and is awaiting review. Both my case manager and the dealership have told me it can take months to be reviewed. The person reviewing it is a regional manager. I've asked for their contact information but so far they've refused to disclose that.
• I've spoken to a lawyer briefly. They seemed to think it wouldn't be worth hiring them for this. They advised filing a claim against the dealership's insurer. As a backup, file a claim with my insurer and have them go after the dealership's insurer to cover the cost. I'm nervous about going this route because if they're not able to recoup then that claim would stay on my record which does have financial implications.
• I've asked the dealership directly for their insurance to file a claim. They refused to give it to me, saying only they file claims against their insurance. I don't know how else to try extracting this information. Cold calling as a prospective customer? Is it hanging on one of their walls maybe?
• I've spoken with the dealership's manager who oversees used cars. They made me an insultingly low offer yesterday, about 10% of its value. For reference, my car is less than 10 years old. Part of the reason it's so low is because they're budgeting in the repair. The manager doesn't seem to care that they damaged my car. But it does seem like they want the car off the lot, so maybe if I hold the line on this front the offer will increase. Unless they tow it, which I do slightly fear.
• I made some phone calls to governmental entities in my state. I've narrowed in on one entity that I'm told should have records of the dealership's insurance and also allow me to file a complaint. I got a voicemail today, but fingers crossed I can connect with them tomorrow.
• I sent an email to an executive last week. If I don't hear back I plan on doing this weekly with different executives. I'll add the names of each executive I've contacted to the bottom so the person has that additional context.
• I've reached out a local TV station who investigates things in the community. If I don't hear back I plan on contacting another one next week. I live in big city so I'm not confident this avenue will be fruitful.
• I've looked briefly into small claims court, but even a positive outcome requires me to collect. So that doesn't seem particularly helpful.
• I've looked briefly into leaving a BBB review, but I'm not sure that will accomplish much.