It follows from the following fact: for any Markov chain its trace on a recurrent subset has the same space of bounded harmonic functions (equivalently, the same Poisson boundary). In particular, the Liouville property for bounded harmonic functions is preserved when passing to a recurrent subset. In the context of random walks this observation was first made by Furstenberg, in particular, if $G'$ is a recurrent subgroup for a random walk $(G,\mu)$, then the induced random walk $(G',\mu')$ is Liouville iff the original one is. For a lamplighter group recurrence of the quotient random walk on the base means precisely that the group of configuartions is recurrent. Now the rest of the argument depends on what assumptions you impose on the "lamp group" (the group of values of a single lamp). For instance, if it is abelian, then the group of configurations is also abelian, so that it is Liouville by Blackwell's theorem (which is more often attributed to Choquet and Deny although they did it several years later). This proof is contained in the 1983 Kaimanovich and Vershik paper.
PS If you want to do it by coupling, then take your favorite coupling proof of the Liouville property for general abelian groups and recast it in terms of the ambient chain on the lamplighter group. However, it does not seem to be a natural thing in this situation. The recurrent subset argument I am referring to is totally elementary.