I booked IST-LHR-DEN for a family of three via an "agency" (Amex travel portal using MR points) under United Airlines. We are traveling with a three-year old, hence the obvious need to reserve seats. The flights are exactly 60 days out. I'm not able to reserve seats for the IST-LHR leg, which is operated by Turkish Airlines. As expected, trying to select seats via united.com gives the message "This flight is operated by our partner".

I've had zero luck trying to see/select seats via Turkish Airlines:

• Website: accepts the reservation code (PNR) but then rejects the ticket number since it doesn't start with 235 (ours start with 016 since it was booked under United)
• Mobile app: I've had success with the mobile app for other flights; for this one, it accepts the PNR and ticket numbers; on the screen with the passenger names listed, the seat area shows "Not selectable"; clicking on it says "We are sorry to inform you that we are unable to process your transaction at the moment. Please try again later." I've tried numerous times to no avail.
• Call center: I've spoken with multiple CSRs, all of whom tell me the system doesn't allow us to reserve seats and to contact United since their flight number is what's shown in the system (even though it's operated by TK)
• Regional sales offices: These may as well not exist; I called four different ones and tried every possible extension, yet still was not able to get a hold of a person.

I'm completely at a loss. Worst case is we beg someone once we board to allow our toddler to sit with one of us parents, although this is less than ideal. Are there any other tricks?

Thanks in advance.