Strongest gear configuration
I would tend to recommend some options which you didn't consider yet: using bevel gears, but actually use a combination of two gear on each axle: a 12t one (the thin one, as you say), and behind it, a 20t one (there you can use a fat one, which will probably be better).

Another option is to use a worm screw, which will also allow you to change the direction of the axle, but which makes the axles non coplanar.

But it really depends on a variety of factors. Gear reduction is one (keeping in mind it has implications on torque, but frankly I'm already way over my head here so I won't tackle that aspect); but torque transmission is another, and so is strength/reliability.

For example, the knob wheel that you did mention has some uses, as it practically can't slip, but at the cost of lower torque transmitted (or something). Worm screws again have another particularity which is unique to them in that they can transfer torque only one way, which also has its uses (locking a mechanism in place).

And then don't forget that the gears aren't all there is to say: sticking your gears closer to the beams will help, as will replacing the two separate beams by an actual frame. You can also make sure one of the axles goes all the way to the other side of the frame.

And last thing, if you're going to transfer power to two wheels, don't forget to use a differential between them (not sure the new one is very solid though).