Rita, I may be misunderstanding something, but wouldn't the following work?

The canonical class of your complete intersection is $(-6+2+3)H=-H$ where $H$ is the hyperplane section. Therefore this is a Fano and hence unirational by the Miyaoka-Mori criterion, which is essentially an application of Bend and Break.

Did you want something more elementary? One can also get this by running the mmp on it, but ultimately it boils down to the same thing. Of course, most likely Enriques's proof is more specific to the situation.

EDIT: here is a short, but non-elementary argument to prove that the threefold in question is uniruled:

The canonical class of your complete intersection is $(-6+2+3)H=-H$ where $H$ is the hyperplane section. Therefore this is a Fano and hence uniruled by the Miyaoka-Mori criterion, which is essentially an application of Bend and Break.