KLM cancelled my flight and rebooked me a day early, and is refusing to accommodate. What to do?

Looking for some advice/opinions on this situation.

Traveling from JFK to OPO on June 2. I was booked on a KLM flight (miles ticket through Flying Blue). KLM cancelled the flight. They rebooked me onto a Delta flight but a day BEFORE my original travel date (June 1). I did not consent to this.

My travel companion was on the same original KLM flight (separate booking) and was rebooked onto Delta flight DL114 on June 2 (our original travel date) using miles.

I want to be on the same flight as my companion, DL114 on June 2. KLM is refusing to rebook me onto it. This summarizes the back and forth with them:

1. First they said there was no availability
2. I pointed out my companion literally just booked onto it with miles, and that Delta's own website shows award availability (same fare, flight and cabin).
3. I called Delta directly and they confirmed availability exists and told me KLM would have to make the change.
4. KLM then said there's no availability for *partner* miles tickets specifically
5. Most recently they told me that because I booked with miles, there's nothing they can do and that the availability on the Delta flight on the same date is for cash fares only.

For context, the same seat on DL114 on June 2 is currently available for purchase at $1,700 cash or the points equivalent. This is probably why KLM is reluctant to rebook me, as doing so would cost them $1,700.

As far as I understand, EC 261/2004 applies to all passengers regardless of how the ticket was purchased (cash or miles). KLM cancelled my flight so they are obligated to rebook me under comparable transport conditions at my convenience. Being moved to a day early is not comparable transport conditions and I didn't agree to it.

If no award seats are available, aren't KLM still obligated to rebook me at their own cost?

I've been going back and forth with them for hours, incurring long distance phone charges, and have everything documented including screenshots of available seats with timestamps and the written email responses from KLM.

I've sent formal emails invoking EC 261/2004 Article 8. Might also try and threaten to file with the ILT and US DOT.

My questions:

- Am I correct that the miles booking doesn't affect my EC 261 rights?

- Is KLM actually obligated to book me onto this flight if no award seats are available?

- Has anyone dealt with something similar and how did it play out?

- Any other avenues I haven't tried?

Flight is coming up very soon so any quick advice is appreciated. Thanks.

Note: To clarify, they also said there are no similar alternative options available for June 2. I'm not insisting on this flight purely because I'm traveling with someone, but more because this is the only reasonable alternative that works for our trip.

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Edit again: I am going to try and look for alternative flights they could book me on but there don't seem to be many reasonable options without a ridiculous layover on the same day. When first booking, this and the Delta flight were essentially the only options. They also told me this in the most recent email follow-up:

"If traveling specifically on June 2 on flight DL144 is essential for you, you may consider the following option:

Request a full refund of your current mileage ticket, and

Purchase a new ticket for that flight through sales channels (either via our website or directly with Delta using a credit card)

Afterward, you may submit a compensation request related to the involuntary change that affected your original booking."

Would this actually work or could I still get screwed over if I decided to do this?