I have an overwhelming need to point out that lime sulfur is a mediocre wood preservative. Modern wood hardeners such a PC Petrifier are much much better at the task. So too is super glue. The downside of these modern remedies is that the wood cannot be recolored after they have been applied.
So, don't apply them until you have produced the coloration of the deadwood that you want (with LS, stains, or whatever), which is, in a sense, the answer to your question - as often as it takes to maintain the coloration you want.