Other question is: is this ridiculous for a bad battery, to just tinker with my car for weeks on end?
The NY Lemon Law kicks in after 30 days, but I suspect all I would get for a leased car would be released (sorry) from the contract, rather than a replacement car or a real solution. And honestly I don't want this to go on another 10 days anyway.
Seem ridiculous to me, if it truly is a bad battery. But I took my car in 3 times to have a wire unplugged for a TSB that should document the procedure in detail and the technician couldn't, so... nothing surprises me now.
Using the principal of "Getting to Yes", what you need to consider is your BATNA (Best alternative to a negotiated agreement). Based on what you said, it sounds like in 10 days, you can invoke the lemon law, stop payments, hand them the keys and walk OR you could wait until they fix the car. These two options are your BATNA.
I would verify that with the lemon law, I could walk from the payments in 10 days without penalty. If I have confirmed that I could actually do that, this is what I would do:
I would then contact both the dealership and Nissan corporate and inform them that I'm displeased with the length of the repair. I would tell them that I'm paying monthly for the car, and through no fault of my own, the car had a failure. I would inform them that it's been 20 days and I've struggled to get reliable transportation at that time, due to the loaner policy and their conflict on who pays for the loaner.
I would then explain that my primary goal is to get my car back in working condition that has been diagnosed and repaired correctly and to have reliable transportation until that occurs. I would inform them that repeatedly negotiating for 5 more days on a loaner is not what I'm interested in--I want reliable transportation until my car is fixed. If they were willing to provide me a reliable loaner until the car was repaired, I would happily wait another <insert time here>, if necessary, to give them time to schedule and do the repair properly. Then see what they offer.
If they are unwilling to negotiate and provide me with those asks, I would inform them that, given they've refused to give me reliable, hassle free transportation, I plan to invoke the NY lemon law and return the car in 10 days, and I will find a new car. The threat that they're getting your car back is likely to get them to start negotiating and get your car fixed fast or reliable transportation. They don't want the car back -- the car has depreciated far faster than the lease payments in the first year of a 3 year lease, so this is a very bad deal for them.
If they refuse to negotiate even with the lemon law on the table, you're back to your BATNA. Either they'll fix your car within 10 days or you'll get a new car.
Finally, because I believe strongly in ethics, I have to do a morality check on my plan. Do I feel that I'm ripping off Nissan by doing this? In my opinion, I feel that you've given them more than enough time to do that repair and you're paying monthly for reliable transportation that you're not getting, so I don't feel that you're doing anything wrong invoking the lemon law in this case. If they had a reasonable explanation about why the repair was taking so long and they were doing their best to serve me, then I would have an ethical concern and would not use the lemon law, but I didn't hear that in your statement.
If you do invoke the lemon law and get out of the lease contract, you can go somewhere else and get a cheap used Leaf, 2018 Leaf (moar range!), or another car if you want.
Note: I am not a Lawyer and this is not legal advice. I have no knowledge as to what the New York lemon law says.