There is no API to query that information as this would require an additional explicit callback. You can use an incumbent or info callback to keep track of the current best known incumbent (which at some point in time will be your MIP start).
If your MIP starts sets all variables then why not just compute the objective function value yourself? That should be pretty easy by just plugging the MIP start values into the objective function.
#CPLEXOptimizers#DecisionOptimization