I noticed one particular gear causing a lot of slipping, the large one driving the bogie. It misbehaved in two ways:
The large gear was pushed sideways by the force of the motor, pushing apart the frame and causing the top gear to slip.
The large gear exerted force on the bogie, pushing one wheel off the ground under force.
To solve these issues, I used a smaller gear and reinforced the frame. This helps keep all the gears together and all the wheels on the ground. The robot can now climb much bigger obstacles.