Can I upgrade to a dimmer switch in my bathroom without using the ground conductor?
The ground, presuming it is actually electrically the grounding conductor and not being misused (illegally) as a neutral, should be pigtailed and connected to both switches.

However: it is difficult to see what the actual connections on the second switch (black/bare) are. If it is a functioning switch and only has the black and bare wires connected to it, then the bare wire must be a live wire in the switch loop. If this is true, then it is both illegal and dangerous to humans and must be fixed.

The fix depends on what is really going on in this box, which is still difficult to tell from the pictures.

You could, for instance, have a single circuit supplying switch loop current on one wire and feeding two different loads through the other two wires. This would be fine. You would just pigtail the incoming live wire to both switches, and pigtail the ground to both switches and to the box.

There are too many unknowns and we really do need to see better pictures of the switches, and pictures of the wiring on the other end (in the ceiling box[es]).