A spiritual weapon (created by the spell of the same name) uses your bonus action for movement and attacking. It does not have its own actions, bonus actions or reactions.
The Spiritual Weapon spell has a casting time of 1 bonus action and range of 60 feet. Breaking down the spell description:
You create a floating, spectral weapon within range ...
So, you use your cleric's bonus action to cast the spell, creating the spiritual weapon within 60 feet of your cleric.
When you cast the spell, you can make a melee spell attack against a creature within 5 feet of the weapon.
If a creature is within 5 feet of where you created the spiritual weapon, as part of the same bonus action you used to cast the spell, you can attack that creature with the spiritual weapon.
As a bonus action on your turn, you can move the weapon up to 20 feet and repeat the attack against a creature within 5 feet of it.
In every round after you cast the spell, for its duration, you can use your bonus action to move the spiritual weapon up to 20 feet and attack a creature within 5 feet of the spiritual weapon.
Since any use of the spiritual weapon spends your bonus action, your cleric still has a regular action (and reaction) available.
It's a DM ruling, but I'd say no.
As you show, nothing in the text of the spell discusses whether it's a valid helper for sneak attack. Therefore, we have to look at other factors.
The Sneak Attack text says,
you know how to strike subtly and exploit a foe's distraction.
But is a spiritual weapon distracting? Note that it doesn't do anything unless ordered by its caster. In particular, it can't make opportunity attacks. Consider the text for opportunity attacks:
In a fight, everyone is constantly watching for enemies to drop their guard. You can rarely move heedlessly past your foes without putting yourself in danger; doing so provokes an opportunity attack.
However, you can move heedlessly past a spiritual weapon, because it's not on guard and looking for an advantage. In the absence of any other text, it's simply floating there until commanded. Thus, on your turn, it's not posing an immediate threat the way another creature would.
Additionally, Sage Advice says no, though without an explanation.
Best Answer
As long as all the rules of spiritual weapon are observed, you are simply reflavoring the spell for fun - no harm done.
But to be clear, you must understand that this has no interaction whatsoever with mechanics such as two-weapon fighting (which means you can add your spell casting ability modifier to the damage!) or anything that says "melee weapon attack", you are still making a melee spell attack.
The other thing you must be aware of is how the spiritual weapon moves in relation to you. If you are moving and want to keep it with you, you must expend your bonus action to carry it with you, and it can only move 20 feet.
Also, you must be aware that spiritual weapon and shadow blade both have a casting time of one bonus action, so it would take two turns to set up your combo, after which you may use your action to attack with your shadow blade and your bonus action to move and attack with your spiritual weapon.
Now if you really want to commit to this particular style (which sounds like a lot of fun!), with a Dexterity score of 13 or higher, you can take the feat Defensive Duelist:
Shadow blade meets the conditions of this feat (emphasis mine):
While wielding your shadow blade, you would be able to use your reaction to add your proficiency bonus to your AC, potentially causing an attack to miss, which would help preserve your concentration on shadow blade.