how to get from Burgas to Istanbul with a family of 5 people? Uber? Cheap taxi?
I doubt there are many taxis willing to cross national borders; a bus might be a better option.
Taxis from Greece take shoppers to Keşan (TR) quite regularly, but I've never seen one from Bulgaria. The buses from there are very decent - run by Turkish companies of course.
Joanna T, Metro Turizm has two buses, a daytime and overnight bus, from Burgas (Bulgaria) to ıstanbul Main Otogar. Have a look: "https:/…". It is best to take them.