You have written so much information without identifying it that I have trouble knowing what you intended to write.
H2CO3 does have two ionization constants, both k1 and k2 but that isn't what the original question asks.
You have the Ka for HCO3^- ok
That is as you have it written, both for the ionization and for the Ka expression.
For HCO3^- as a base, it should be
HCO3^- + H^+ ==> H2CO3
or if the hydolysis equation in which HCO3^- acts as a base it is
HCO3^- + HOH ==> H2CO3 + OH^-
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