Carbon Dioxide levels appear to be linked to the Milankovitch cycles according to Ice Core data but what is the mechanism?
The link between orbital parameters and CO2 levels is, as you wrote, the temperature. 

Orbital parameters may change, via feedback mechanisms, the atmospheric temperature. To my knowledge the most important feedback mechanism with respect to the Milankovich cycles is the ice-albedo feedback. 

The temperature, however, is an important player in geochemical feedbacks which influence the concentration of gases like CO2 or methane. For CO2, again to my knowledge, you named the most important feedback; the exchange of CO2 between atmosphere and ocean. 

The exchange of CO2 between ocean and atmosphere in turn depends on the partial pressure of CO2 in the atmosphere and the surface layer of the ocean; there is uptake, release according to the pressure gradient.
Now, the partial pressure of CO2 in the surface layer depends also on the amount of carbon transported or received from the deep sea layers. There is way more CO2 stored in the cold water of the deep sea. So the exchange of CO2 does depend on the vertical circulation pattern of the ocean.

There are, to my knowledge, three major mechanisms that transport carbon between the surface and the deep sea and they all depend on temperature.

Solubility pump: Cold water can store more CO2 than warm water. Therefore at the poles, where cold water is sinking into the deep sea, you have CO2 rich water. The CO2 stored in the deep sea can, as far as I know, remain there for hundreds of years, before it comes up again in the warmer upwelling regions at the equator. The temperature dependence of that mechanism is obvious.

Biological pump: In the surface layer you have photosynthesis of algeas etc., so carbon is withdrawn from the layer and stored in biomass. A portion of dead plants is not reused (eaten) but sinks into the deep sea. The process is temperature dependend since biological activity depends on temperature.

Carbonate pump: The amount of carbonate and CO2 are linked to each other via marine chemistry. Animals building shells etc. need carbonate thereby also influencing the amount of CO2 present in the water. Again, a portion of this carbonate is withdrawn from the short term cycles once the animals die and sink to the deep sea. Again this mechanism is temperature dependend since biological activity is.