Thank you for the clarification! I am not a biochemist, so I am not sure that I can give you good advice in this matter. However, to me this looks like a case not of “melting”, but a chemical reaction. If the circular dichroism changes during the reaction, you should be able to follow the reaction. If the separation of the DNA strands is a reversible chemical reaction with no kinetic hindrance, you should be able to obtain something like a reaction enthalpy; if the reaction is significantly controlled by kinetics, you would extract the activation energy.