CAn a jump starter power an trailer slide out?

Unless you have a very unique braking set up your brakes are powered by the electric controller through the trailer plug.

If the TT becomes separated from the TV and the plug is disconnected, the separate brake switch cable stays with the TV. This activates the emergency switch and the brakes are now energized by the TT battery and lock up.

That is how it should work, whether it does or not depends on you.