The switch is part of the panel. The same Panel that has the park break.
Since the light switch is part of the instrument panel, the only voltage coming from the instrument panel and going to the light circuit is to energize the relays. This should not cause a significant draw on voltage.
The instrument panel receives voltage from serveral sources. You always have power to the panel at pin 3. When you turn the ign switch on you have power to pins 23 and 24. Since the lights only work with the key on, I would assume these are the pins providing power for the light circuit.
I am thinking the light relays are in front of the cab fuse panel. The headliner needs to be removed to see them. I would remove the two relays and try the light switch again and see if you receive the same result. Also with the headliner removed check the ground location. It is either in front of the cab fuse panel or behind it, and may be one or two bolts welded to the cab.
You can also check for a voltage drop at the 15A fuse in the cab fuse panel. It should be the second fuse from the front. The panel receives the key on voltage from this fuse. Remove the fuse and check the connectors in the block so they are in place and not damaged or the ears spread to far apart. Replace the fuse and using a voltmeter probe one side of the fuse while turning on the lights and see if the voltage drops. You may need to remove the fuse block and probe the back of the wires just to make sure the fuse is making good contact.
A loose wire on the ignition switch can also cause a voltage drop in this circuit, though I would think you would see other symptons too. Still check it.
If you have a significant voltage drop at the fuse, then we need to look further upstream from the fuse.
You may also have an internal short in the instrument panel. If everything else checks out you might see if a friend has a loader with the same panel and try it in your machine.