The impact of any read-through from a leaky stop codon in an expression unit with only one CDS would probably depend on a few things, mainly (i) where is the next in-frame stop codon, (ii) what are you trying to express, and (iii) how leaky is the stop codon?
In cases where the next in-frame stop codon is only a few base pairs away, there would probably be little impact, however in other cases the next stop codon could be far away. In these cases, there are two things which may cause an impact.
The first is that a long peptide sequence could be added to your protein, which depending on what you are expressing, may cause your protein to misfold or lose functionality.
The second is that you could get ribosome stalling, especially if any of the codons between your stop codon and a second stop codon require rare tRNAs. As you mentioned in your question, there are mechanisms for rescue in these scenarios, however if your CDS is expressed under a strong promoter on a high copy number plasmid, this mechanism may need to be mounted much more often than usual and cause burden on the cell. I should note that this is speculation on my part as I can’t find any studies which have shown this.
The actual impact any of these scenarios may have on your system would likely be strongly dependent on how leaky the stop codon actually is. As there are many examples of constructs which use only a singe TAA in their design with no apparent negative affects, presumably under ‘normal’ circumstances there is little risk to using a single stop codon.