This is an interesting idea, but it's highly unlikely that the Order of St. Lazarus bears any relation to the Cagots.

The Cagots or Agotes appear in medieval sources in the Pyrenees region--south/western France and into Spain. The interesting thing is that the restrictions evident against them in later sources do line up with medieval persecution of lepers! Of course, what was "a leper" in the Middle Ages? In R.I. Moore's famous formulation, a leper was anyone who other people labeled as such. So it's quite possible that whatever cultural/social group the Cagots may have been beforehand got stamped with the mark of "leper" at the time when Europe was apparently facing an epidemic of leprosy and certainly (re)inventing major persecution of lepers.

The Order of St. Lazarus after 1291 is afforded a central base in Europe in Poigny, which is in north-central France. At that time, the French crown is working on expanding its control of "French" territory outwards, but the political connections would still be rather tenuous. The reason reason I would argue against any overlap is social status. The Lazarites, however their role is evolving in the post-Crusader state era (most importantly, losing their hospitaller role!), have a certain amount of wealth and social prestige. The Cagots are the opposite.

So while there may very well be some connection between the Cagots and the social idea of leprosy (medical disease notwithstanding), they are almost certainly not the descendants of a community of Lazarite knights.