Are 2A cleavage sites functional in bacteria?
2A self cleaving peptides do apparently work in bacteria (as the papers in your question suggest), but it seems that they are not commonly used in these chassis simply because they are not needed.
For context, in prokaryotes, translation is initiated by the ribosome binding to a shine-dalgarno (SD) sequence on the mRNA. These SDs can be placed anywhere along the mRNA strand, and so multiple SDs can be present. This means that translation can begin at multiple points along the mRNA, and hence proteins can be co-expressed by one promoter by including multiple RBSs and stop codons within the expression unit (i.e. promoter-RBS-CDS1-stop-RBS-CDS2-stop-terminator).
However, in many eukaryotes, translation begins at the 5' end of the mRNA where the ribosome is assembled, and scans along the mRNA until it finds the start codon. Once the protein has been synthesised and the stop codon reached, the ribosome falls off of the mRNA. This means that if there was another CDS encoded after the first stop codon, it won't be translated.
It should be noted that it is possible for eukaryotes to generate mRNA encoding multiple proteins, but as it was assumed for a long time that this was unique to prokaryotes, there is less research for this in eukaryotes. This paper has some useful information: https://academic.oup.com/femsyr/article/2/2/215/536601.
Because of these differences, for most applications, if co-expression in bacteria is required it is much simpler to add in multiple RBSs and stop codons than to rely on something like 2A peptides which, as you mentioned, do not always work and can result in fused proteins. However in eukaryotes, techniques such as 2A peptides are important considerations for co-expression as poly-cistronic mRNA (mRNA encoding multiple proteins) seem to be much less common.
This doesn't mean that there aren't applications for 2A peptides in bacteria, but they are much more specific and hence their use doesn't turn up so much in the literature.
These wikipedia pages have some good reading and references:

https://en.wikipedia.org/wiki/Internal_ribosome_entry_site
https://en.wikipedia.org/wiki/Eukaryotic_translation

Disclaimer: I don't work with eukaryotes so most of the information about gene co-expression and how common it is has been gathered from reading for this question.