the mantle is in a highly reduced state
This is not entirely correct. The core is in a highly reduced state, but the mantle is not necessarily reduced, and is quite oxidised in some places. The mantle is heterogeneous.
however I justify its emission because carbon dioxide is continually being fed into the mantle by the subduction of carbonate sediments, is this correct?
The carbon dioxide in the mantle does not necessarily derive from subduction of carbonates.
Or is this all wrong, and the gases are being emitted because they are still being degassed from the solid mantle in the process if heterogenous separation which separated the atmosphere from the mantle. If this is the case, why have all other gases been separated but not the two I have mentioned?
Not entirely accurate. Although it is true that some of the gases in the atmosphere derived from the differentiation of the core-mantle-crust system, a popular theory suggests that many of the gases came after the differentiation through other planetary bodies such as comets and meteorites. You are also implying that carbon dioxide and sulphur dioxide are the only two gases that are emitted from volcanoes. No. Water vapour is also emitted (commonly much more than the other two) and also other gases. It's just that less people care about the water and it gets less publicity, because, well, it's just water. There are other gases that are emitted as well: nitrogen, the noble gases. But they are much less common than the other two. They are not the only gases.
The authors of Fluxes and sources of volatiles discharged from Kudryavy, a subduction zone volcano, Kurile Islands write:
Several potential sources for CO2 can be identified: (1) hotspot-type mantle, (2) MORB-type mantle, (3) subducted oceanic crust (MORB), (4) marine carbonate, (5) organic C from crustal and subducted sediments
They also give a table which states that in island arc volcanoes, 12% of carbon is sourced from the mantle, 67% is sourced from carbonates and 21% is organic carbon. Note that this only gives the amount degassed from volcanoes in subduction zone arcs. This does not mean that all carbonate-hosted carbon is released from the rocks to be degassed. I read somewhere that only 50% is degassed, and the rest is lost to the deep mantle. I would also expect that MORB magmas will have significantly less carbonate-sourced carbon.
Also, The deep carbon cycle and melting in Earth's interior
may be of interest.