No compensation under EU261 since origin+destination is outside of the EU. Even if you fly with an EU carrier, you must either start or end your trip in the EU.
It still does not hurt to try: https://www.lufthansa.com/us/en/feedback

File for an involuntary full refund with CapitalOne.
If they say, they cant process a refund because they have no control over the ticket anymore, tell them, yes, they might not be able to process the refund in GDS, but they can file a refund application via "bsp link"

Now asking whose fault is it here?
Yes, airlines claim its IATA standard that a TA booking is 'owned' by a TA and therefore, the TA has to take care of the customer.
So in case of an ATC (airline initiated change = schedule change, cancellation), the TA has to process the change.
However, if the passenger does a voluntary rebooking directly with the airline, the airline is obligated to do the full reissue / exchange.
With most airlines TAs only have read access after the airline change something in the eticket record, so even if they wanted to reissue, they simply cant anymore since they don't have write access anymore.
So clearly, its fullys LH fault here.

Chose a different TA. I would advise not to go with an OTA and also not book directly with the airline since you get the same bad service. Instead, book with a TA where you pay like 25$ for the service.