Algorithm for 3-object matching
It sounds like you're looking for a polynomial-time approximation algorithm for the 3-dimensional matching problem (which is, as you noted, NP-complete - the decision problem is one of Karp's 21 NP-complete problems).

In 2014, Ostrovsky and Rosenbaum proved that 3D matching is APX-complete. This implies the existence of an efficient algorithm that produces a result within a "fixed multiplicative factor of the optimal answer" - in their case, $4/9$.