First check http://pubs.acs.org/doi/abs/10.1021/je060430q. The mixture K2CO3 + CO2 is equivalent to a mixture of K(HCO3)2 and K2CO3.
Then search for the Setschenow contant of CO2 for aqueous KHCO3 in the old books on solubility. If you don't find it, take the one for Na2HCO3 from http://pubs.acs.org/doi/abs/10.1021/je1011168, and use some model of the Hofmeister effect to recalculate the Setschenow constant of a potassium salt from the value of the sodium salt.
If you are wanting to review the experimental carbon dioxide solubility, you may want to look at Kamps et al., Solubility of CO2 in Aqueous Solutions of KCl and in Aqueous Solutions of K2CO3, http://pubs.acs.org/doi/abs/10.1021/je060430q. Stolaroff et al. also has some information, https://e-reports-ext.llnl.gov/pdf/764798.pdf.
I don't know if you want to measure it yourself experimentally ....
I do not have the setup to measure it experimentally. I have the 0.5 M solution of KHCO3, CO2 is bubbled through it until saturation. What I want is to find the concentration of CO2 after the solution is saturated. I read the papers but I could not find data about this information. I understand the system is somehow complex because of the reactive nature of CO2 dissolution in water.