To get started I borrowed a method that worked well for the "how many ways can 3 sticks touch" problem and considered numbers of intersections. With 0 intersections we have a version of the problem of how many ways 3 bags can be stuffed that @samjshah wrote about (https://samjshah.com/2015/04/03/stuffing-sacks/).
If there is just one intersection it must come from two circles kissing. There are two ways the circles can kiss. Each row below shows the possible placements for the third circle for one of the types of kissing.
4 intersections could come from two crossings, three crossings that share an intersection, or one crossing and two kisses:
https://www.cake.co/conversations/lQlZDGj/how-many-ways-can-3-circles-intersect-how-many-ways-can-n-circles-intersect>, especially Factotum's numbering system. I'm sure a few of the above possibilities came from Factotum's enumeration.