How can I repair a buried 120v cable splice to make it meet code?
What you call "Romex" is actually NM cable.  It is not rated for outdoor,  wet locations nor direct burial.  UF is rated for all three.

For direct burial rating,  what matters is the lettering on the cable, not the color.  Several vendors are choosing to use white for NM 14, yellow for NM 12, orange for NM 10 and  gray for UF of any size. But that is not a standard and has no force of law 

Copper is not a legal conduit wiring method, so it is irrelevant and it counts as direct burial.  

Direct burial requires 24" burial depth. Rigid conduit requires only 6". 

Direct burial splices are not allowed except with certain splice kits designed specifically for that.

This would be a hot mess if it were mains wiring. 

But it's low voltage DC. Right?

Because these defects would not be of concern if this circuit were less than 30 volts and less than 55 watts.  

So I would be thrilled to hear that the cable actually goes from the switch to a low voltage power supply  in the 12-24V range, which then powers 12-24V LED lights at the pole. These are readily available for RV or automotive use. 

I presume it might be low voltage since it can easily handle the load, so there's no reason to risk electrifying the ground and wasting energy from AC mains  leakage at the splice