No, if the signifiers and signified are defined by direction of causality (fire causes smoke, signified causes signifier), then by the order of cause the amino acid residues in a protein sequence are the signifier and the codon the signified.
They certainly aren’t symbols because they’re not defined by culture, tradition, or convention.
So if tree rings can’t be considered a code because they’re indexes, then neither can the genetic code, because it would be defined by a direction of causal relation between signifier and signified, not by culture, tradition, or convention.
There is nothing you can do against this basic fact.