I have a unit at Battlefield A at start of my turn. I score a point for holding. I move that unit to base and lose control of the battlefield. Then, I move Kaisa Survivor from base to that same battlefield. I know I don't score a point, but do I technically conquer? Do I draw a card from Kaisa's ability?

No, you do not conquer the battlefield, and you do not draw a card from Kai'Sa, Survivor's ability.

According to Rule 465 and Rule 466.2.c, a player may only score each battlefield once per turn, regardless of the method (Hold or Conquer). Because you already scored Battlefield A by holding it at the start of your turn, you have exhausted your scoring opportunity for that battlefield for the remainder of the turn.

As established in Rule 446.1 and confirmed by multiple FAQs (such as FAQ #3165 and FAQ #8785):

• Conquering is defined as gaining control of a battlefield you have not yet scored this turn.
• Since you have already scored Battlefield A, moving a unit back to it does not qualify as a "Conquer" action.
• Because you are not performing a "Conquer," the "When I conquer" triggered ability on Kai'Sa, Survivor does not trigger.

Even though you retake control of the battlefield, it does not count as a conquer, and therefore no conquer-related effects will activate.