Hello everyone, I'd like to update this post with my experience should you come across this issue in the future.
We arrived in TLV airport on a separate Turkish Airlines TLV-IST, paid for using Chase UR miles. I asked the agent if she can checked-thru our luggage to our IST-MIA flight. I showed her the confirmation for the second segment. She said no problem.
I also asked for exit row seats for both flights which was granted at no additional charge.
She asked to look at our carry-ons and said only one carry-on per person is allowed. I told her the backpack is a personal item and the rollaboard is our carry-on. She responded that "the flight is full" and offered to check our rollaboard at no additional charge.
Considering we had separate itinerary and that the allowance for TLV-IST is only one check-in luggage at 20kg while IST-MIA is two luggage at 23kg each, we're grateful of this offer.
The flight experience was good as we enjoyed the meals, seats were comfortable and the in-flight service was good. We had a young male FA as our attendant and he's only been with TK for two years.