We show that the groups mentioned in the title are solvable. Moreover, a point stabilizer H of the locally finite barely transitive group G is almost nilpotent, whenever the indices H : H \cap H^g (g \in G) have a finite upper bound.
KUZUCUOĞLU, MAHMUT (1997) "Locally Finite Barely Transitive Permutation Group with Almost Nilpotent Point Stabilizers," Turkish Journal of Mathematics: Vol. 21: No. 5, Article 8. Available at: https://journals.tubitak.gov.tr/math/vol21/iss5/8