Possibly but I don't think Stormbreaker has the ability to keep the Bifrost open to be able to destroy the ship. We see in Thor that it is the act of leaving the Bifrost open which could have destroyed Jotunheim and not having used the Bifrost to "cut through" the planet.
Volstagg: Couldn't you just leave the bridge open for us?
Heimdall: To keep this bridge open would unleash the full power of the Bifrost and destroy Jotunheim with you upon it.
Stormbreaker, however, only seems to be able to summon the Bifrost so it would appear that it probably wouldn't be able to keep it open.
Eitri: It's a mold. A king's weapon. Meant to be the greatest in Asgard. In theory, it could even summon the Bifrost.
The only time I can think of when we see the Bifrost cut through something is at the beginning of Thor: Ragnarok when it cuts off the dragons head. However, the dragons head was in the "portal" type area for being transferred and its body was not.
It's unclear if the Bifrost would cut through anything in its path. In fact if anything it seems it would "dodge" what is in its path as we see at the start of Avengers: Infinity War when Heimdall sends Hulk back to Earth. The path it takes is not direct but curves a bit dodging planets. Also when Heimdall does this the Bifrost only sends Hulk along the path and doesn't rip a hole in the ship they're on.