Reporting Higher Score in LMS

This is something that really needs to be more managed by your LMS.  There are two factors in play here:

• How many attempts you are giving the learner in the Captivate quiz DURING A LAUNCH SESSION.
• Whether or not your LMS has the facility of deciding which of several attempts ON THE ENTIRE MODULE it will use for reporting.

In Captivate, if your learner gets to the Quiz Results slide but has not as yet passed the quiz, you can give them the option to Retake Quiz and try for a better score.  If the learner clicks that Retake Quiz button, their current score is NOT sent to the LMS.  It would however be send to the LMS if they clicked the Continue button, allowing the Quiz Result slide timeline to continue playing to its end.  This is why you are only seeing the last reported score in your LMS. The LMS is only getting one score per user session.

If you set the Captivate quiz to only allow ONE attempt, then there will be no Retake Quiz option.  To reattempt the quiz and try again, the learner would need to relaunch the entire module from the LMS.  This way, each launch of the module will result in a different score being sent to the LMS.

THEN the next issue is how your client's LMS decides to retain or display the scores it receives from course module launch sessions.  Some LMSs (Moodle for example) give you several options.  You can keep the last score, highest score, or average of all scores received.  But Moodle (ironically) does this bit a lot better than many other much more costly LMSs. 

Since I don't know what the capabilities of your client's LMS are, this is something YOU will need to find out.  And I suggest you find out FIRST before wasting time trying to reconfigure your Captivate module to do something that might not be possible with that LMS anyway.