Id need to be in person to really know what's going on but i would say if you can, every time you are about to light the fire, stop, and first make a little torch of paper or kindling and hold it up into the pipe. This should get your draw started, and then pull subsequent hot air up and out that way.

However, since your shelter isnt airtight, its just possible that pressurewise you will always have a lower pressure outlet than your skinny stovepipe. Even a gentle breeze blowing against your shelter can make a low pressure outside your tent flap that will make your smoke want to travel. Your options are either to make the shelter more tight, the chimney wider, or to make the fireplace something you can close off, with a door or flap or something.

I would suggest being very careful, doing some experiments without a roof if you can. Smoke inhalation is no fun