There is no reason to assume they are indifferent. By attacking the Cloud Tribes they can weaken the nascent Cai-Bai alliance and gain some good reputation. Starting a war with the Polar Nations does the opposite.
You're thinking from the perspective of someone who values peace. Sun Shao does not. His mindset is dominated by a highly specific in-group vs out-group social filter combined with his every action being dominated by conflict and death as a driving force.
Attacking the Polar Nations is substantially little different from attacking the Cloud Tribes save in scale, and that's even more pronounced because the Emerald Seas would be dragged into allying with him by default due to the shared border with those powers
If Sun Shao can get away with it - and as a mobile, active White with guaranteed White level allies he absolutely could - then there is nothing
but benefits from his perspective.
Conquerors are voracious, eternally starving creatures. Whether it's the Jungle or any other acceptable target, the perspective doesn't change. Only the means of succeeding or not.
And frankly the Thousand Princes would likely beat his ass like a drum if he were to try and invade, by virtue of forcing them to unite against him. And being the only Imperial Province bordering their territories. Stating his reluctance to drive west is indicative of him lacking drive is false. Sun Shao picks fights he can win and
then commits to them. He would not be able to fight a peer polity to the Empire alone and thus does not.
Playing the hammer to the Cai clans Anvil is far far more manageable, especially with the various Sects having a number of Prisms available and invested in any such conflicts willing to back him should things come to that.
Would it mess up the Emerald Seas and ruin their ability to recover?
Sure.
But why would he give a shit about Cai Shenhuas well being? Or that of the Emerald Seas as a whole?
Fuck 'em.
Edit: Hell, theres tons of ways that forcing Shenhua to face tank the Polar nations at large while Sun Shao himself takes most of the glory serves to his active advantage. Not the least of which is completely fucking over one of the Bais direct allies amongst the Provinces, thereby weakening the Thousand Lakes own ability to recover their economic strength or set up their own port - a direct rival to one of Shaos own previously failed initiatives.
If Sun Shao successfully drags the Polar Nations into a war, hes absolutely the type to play it to his advantage and feel zero remorse for doing so. If anything he'd be smirking.
Gotta remember that "Protect Family; Fuck the Bai" are the overriding political goals of the Sun. Fucking over the Cai fucks over the Bai as well. It also fucks over the Xuan which bolsters the Jin - another Sun ally - and strengthens the maritime trade route that Sun Shao would be trying to set up (Jin ships to Western Jungle port) which in turn strengthens them both in a virtuous cycle to the expense of the Bai while helping solidify Sun Shaos own domestic position and letting the Jin launch more raids out west with impunity.
Its cascading plusses on a theater-class scale.