What is the use/function of lithium chloride as an excipient in sofradex eardrops?
Probably the main reason $\ce{LiCl}$ is used in liquid to soften ear wax is that $\ce{LiCl}$ is quite hygroscopic and pharmacologicically inactive under the conditions used (small amount used to retain moisture).

As you state, taken internally, in contact with the eyes, or left in contact with skin for extended time, there are some biological effects, but I'd not be concerned about those with a tested commercial product.