Well, technically, the type of
Alectoris is "
Perdix petrosa" by monotypy, this being a recombination (originally by
Latham 1790) of
Tetrao petrosus Gmelin 1789 (now in
Ptilophachus), here misapplied by Kaup to another species with 'feathery' tail, with males getting small bumps on the tarsi during the mating season, and said to live in the mountains of southern Europe with habits similar to those of
Perdix saxatilis (=
P. saxatilis Bechstein 1805, a syn. of
Alectoris graeca (Meisner 1804)).
Kaup 1829:
https://biodiversitylibrary.org/page/41576603
The conventionally accepted taxonomic identity of the type species is based on the assumption that Kaup was using
Perdix petrosa (Gmelin 1789)
sensu Temminck 1815, as was usual in his time.
The adjunction of something like "=
Perdix barbara Bonnaterre" by a subsequent writer (here Peters 1934) to a type species name is taxonomy, not nomenclature. This merely indicates the valid name of the taxonomic (sub)species that said writer recognised as including the nominal type species. The type species itself remains unchanged, being, exclusively, what is left of the '=' sign; what is to the right of this sign is determined subjectively and, in principle, subject to permanent taxonomic reassessment. (I personally prefer not to use '=' signs at all for subjective synonymy statements, because I feel it is deeply misleading; but it's a very frequent practice, which is hard to ignore.) In the present case, the identity of the type specimens of Bonnaterre's name should play no role whatsoever in how the generic name
Alectoris is applied; this should in principle be entirely determined by the ID of the types of
Perdix petrosa.
(Note also that, in principle,
Alectoris should not be used at all --
Caccabis, which was universally used for this group before
Hartert 1917 claimed 'page precedence' for
Alectoris, has priority by first reviser action of
Gray, in 1846.)