I'm not sure what you mean by "don't have these amino acids in their DNA structure". There are 64 codons that code for amino acids and start/stop signals. But there are only 20 common amino acids, so multiple codons typically code for the same amino acid. However, just exactly what happens depends on the set of tRNAs that are being expressed. Plus the enzymes that catalyze attachment of amino acids to tRNAs.
That is, what's in the DNA is codons, and what amino acids get incorporated depends on tRNAs etc, plus what amino acids are around (synthesized or ingested).
Anyway, I get that I was being a little pedantic. But in the paper referenced by TFA, I see that they studied three nonproteinaceous amino acids: ornithine, 2,4-diaminobutyric acid, and 2,3-diaminopropionic acid. Canavanine is much closer to arginine, with a methylene replaced with an ether bridge. I wonder how well it would have polymerized.
0) https://www.pnas.org/content/116/33/16338
Edit: Upon reflection, "there are just 20 proteinaceous amino acids" and "AS/400 is a database" are ~similar simplifications.