Clique-width expressions with logarithmic depth
After a while I found an answer in the literature, so I'm posting it here in case it is useful to someone else.

It is in fact possible to re-balance clique-width expressions so that they have logarithmic depth. The result is given in the paper "Graph operations characterizing rank-width and balanced graph expressions" by Courcelle and Kanté, WG '08. I quote Theorem 4.4 from the paper:

"Every graph of clique-width or NLC-width $k$ is the value of a 3-balanced
clique-width expression of clique-width or NLC-width at most $k \times 2^{k+1}$"

The catch here is that the number of labels blows up exponentially in the balancing. It seems that for clique-width no better result is currently known. The same paper gives a similar result with only a constant blow-up for rankwidth, but this doesn't help, since the difference between clique-width and rankwidth can be exponential in the worst case.