It seems to me that such a manifold exists.

Indeed the small deformations of the "symmetric" Mukai-Umemura $3$-fold $X$ are described explicitly by Donaldson in this paper, pages 43-44. There he describes 5 classes of deformations. Classes 1,2,3 correspond to points in $H^1(X,TX)$ whose $PSL(2,C)$ orbits are closed, and the corresponding small deformations admit Kahler-Einstein metrics thanks to a theorem of Székelyhidi, see Propositions 7,8 and page 12 of this paper.

Cases 4,5 have orbits that are not closed, and do not admit Kahler-Einstein metrics thanks to Tian's argument (case 5 contains the manifold considered by Tian). These are the only possible cases where you could find an example. Any such manifold has a nontrivial C* action precisely if the $PSL(2,C)$-stabilizer of the corresponding point in $H^1(X,TX)$ contains a C*.

It seems to me that in case 4 this stabilizer cannot contain any C* since having such a stabilizer would imply that the polynomial has at most two distinct roots. On the other hand, it seems to me that one could pick an example in case 5 with one root with multiplicity 5 and one root with multiplicity 3 which should have a C* in its stabilizer, and this could be your example (I might be wrong, though).

[Edited] Such a manifold cannot exist.

Indeed the small deformations of the "symmetric" Mukai-Umemura $3$-fold $X$ are described explicitly by Donaldson in this paper, pages 43-44. There he describes 5 classes of deformations. Classes 1,2,3 correspond to points in $H^1(X,TX)$ whose $PSL(2,C)$ orbits are closed, and the corresponding small deformations admit Kahler-Einstein metrics thanks to a theorem of Székelyhidi, see Propositions 7,8 and page 12 of this paper.

Cases 4,5 have orbits that are not closed, and do not admit Kahler-Einstein metrics thanks to Tian's argument (case 5 contains the manifold considered by Tian). These are the only possible cases where you could hope to find an example. Any such manifold has a nontrivial C* action precisely if the $PSL(2,C)$-stabilizer of the corresponding point in $H^1(X,TX)$ contains a C*. However one can check directly that in both cases the stabilizer cannot contain any C*

It seems to me that such a manifold exists.

Indeed the small deformations of the "symmetric" Mukai-Umemura $3$-fold $X$ are described explicitly by Donaldson in this paper, pages 43-44. There he describes 5 classes of deformations. Classes 1,2,3 correspond to points in $H^1(X,TX)$ whose $PSL(2,C)$ orbits are closed, and the corresponding small deformations admit Kahler-Einstein metrics thanks to a theorem of Székelyhidi, see Propositions 7,8 and page 12 of this paper.

Cases 4,5 have orbits that are not closed, and do not admit Kahler-Einstein metrics thanks to Tian's argument (case 5 contains the manifold considered by Tian). These are the only possible cases where you could find an example. Any such manifold has a nontrivial $C$* action precisely if the $PSL(2,C)$-stabilizer of the corresponding point in $H^1(X,TX)$ contains a $C$*.

It seems to me that in case 4 this stabilizer cannot contain any $C$, since having such a stabilizer would imply that the polynomial has at most two distinct roots. On the other hand, it seems to me that one could pick an example in case 5 with one root with multiplicity 5 and one root with multiplicity 3 which should have a $C$ in its stabilizer, and this could be your example (I might be wrong, though).

It seems to me that such a manifold exists.

Indeed the small deformations of the "symmetric" Mukai-Umemura $3$-fold $X$ are described explicitly by Donaldson in this paper, pages 43-44. There he describes 5 classes of deformations. Classes 1,2,3 correspond to points in $H^1(X,TX)$ whose $PSL(2,C)$ orbits are closed, and the corresponding small deformations admit Kahler-Einstein metrics thanks to a theorem of Székelyhidi, see Propositions 7,8 and page 12 of this paper.

Cases 4,5 have orbits that are not closed, and do not admit Kahler-Einstein metrics thanks to Tian's argument (case 5 contains the manifold considered by Tian). These are the only possible cases where you could find an example. Any such manifold has a nontrivial C* action precisely if the $PSL(2,C)$-stabilizer of the corresponding point in $H^1(X,TX)$ contains a C*.

It seems to me that in case 4 this stabilizer cannot contain any C* since having such a stabilizer would imply that the polynomial has at most two distinct roots. On the other hand, it seems to me that one could pick an example in case 5 with one root with multiplicity 5 and one root with multiplicity 3 which should have a C* in its stabilizer, and this could be your example (I might be wrong, though).