Pantothenic acid is a precursor for the synthesis of Coenzyme A which is essential for most, if not all, living organisms. As bacteria appeared on earth before plants, and can currently live in environments where there are no plants, it is difficult to envisage their being unable to synthesize pantothenic acid. The situation with yeasts is less obvious, but as independent free-living single-cell organisms, the expectation would be that they had retained this synthetic ability of their evolutionary precursors.
Pantothenic acid synthesis in different organisms
Pantothenic acid is produced by a wide range of bacteria, with the structure of the enzymes of the pathway of its synthesis, e.g. the final enzyme, pantothenate synthetase, available from many different species, as can be seen from a search of the Protein Data Bank.
Indeed far more detail is known about the process than in higher plants, because of the greater ease of conducting molecular genetics on the former.
One way to check for the existence of a pathway in a particular organism is using KEGG, and this approach shows the pathway for the synthesis of pantothenic acid is present in Saccharomyces cerevisiae.
The poster refers repeatedly to the efficiency of production of pantothenic acid but appears to be misusing the word, and it is not clear in this context what he means. In science efficiency expresses the percentage of input energy or material that is converted to an intended output. For example in biochemical thermodynamics, inefficiency might be the loss of chemical energy as heat (in processes where the aim is not to produce heat). In a general sense such efficiency exerts selective pressure on organisms, and this will be greatest in rapidly-growing organisms such as bacteria which are subject to competition for resources. All organisms have regulatory mechanisms to ensure that there is sufficient of the product of a pathway when it is required, but that energy and precursors are not wasted by producing it when it is not required. Such regulatory mechanisms tend to respond more rapidly to changes in bacteria than in eukaryotes (such as plants), with their longer division times, but there is no reason to believe that any organism that synthesizes pantothenic acid does so inefficiently.
It is possible that the poster is interested in which organisms might present the most abundant sources of pantothenic acid for those organisms such as mammals that cannot synthesize it themselves. This may relate to the size of the ‘pool’ of this metabolite in the cell. However that is a completely distinct question from synthesis and efficiency thereof. I doubt whether there is much literature on this as it appears that human deficiency of pantothenic acid is very rare, so that this has not been a particular dietary concern.