Despite what the Turkish Airlines phone operator may have told you, the simple fact is that the most likely result is that you will be denied boarding of your flight from Turkey to Panama.
Like most airlines, Turkish use the Timatic system at check-in to confirm a passenger meets the requirements to enter their destination country, as well as for any transit countries - and Timatic clearly states that a visa is required for transit of longer than 12 hours.
TWOV (Transit Without Visa):
Passengers with a confirmed onward ticket for a flight to a third country within 12 hours. They must stay in the international transit area of the airport and have documents required for the next destination.
If you were to obtain some form of written exception from the consulate it is possible that the Turkish Airline staff would accept this as valid, but even then there's no guarantee. If Turkish airlines allows you to travel to a country without the necessarily paperwork/arrangements (in this case, a visa or an outbound flight within 12 hours) then they can be fined by the government in the destination country, and may be forced to return you to your initial origin airport, so these regulations are generally taken very seriously - especially in the times of COVID-19.