Yes, on the MO return the 77% is the proportion of MO source income (allocated not including remote work outside of MO) divided by MO adjusted gross income(AGI), which is federal AGI with MO additions/subtractions as if MO resident.
Yes, the KS return should use the MO allocated amount. This typically is automatic in TurboTax however, you can adjust the amount in the "step-by-step mode.
Go back to the KS interview and update the "Credits and Taxes" section. Revisit the "Credit for taxes paid to another state" topic and edit the MO line. On the next page you can allocate the part of wages that are MO sourced (allocated to MO) and the tax liability on that income on the page after that you can adjust the "Total Income" from MO sources if there was MO income other than wages that need to be accounted for. If not, then the box should match the MO wage allocation amount.
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