Again, I think the only problem here is folks not really considering the implications of a world where magic is just as real as physical bodies. As far as I am concerned, the cleric in the OP's example is in range of the fireshield and smacking it with a weapon. The fact that the cleric is hold the weapon with their magic rather than with their hand makes no difference to the magical effect; the cleric is still connected to the weapon and thus takes the fire shield damage if they are within its range.
To me, them not taking the damage would be weird - it would be implying that magic is somehow not as real as physical bodies, when in the game it very much is.
It's partly just how I envision it. The fire shield obviously isn't blasting out in all directions, if it's exceeding the 5 foot area around the creature that cast it, it's not by enough to catch everyone around it, only the person holding the weapon doing the attack.
So let's say A has a longsword, B has fire shield, C has spiritual weapon, D has a polearm with 10 foot reach. If A attacks B with their longsword, A takes damage. I envision it as a combination of things, A has to physically encroach on B's space in order to attack, their arm is entering B's square. It could just be the arm that is set on fire or it could be that the arm (or even just the weapon) touches the fire and a flare of fire encases A following A's arm up the body. If D attacks B with their polearm, the fire flares out but no part of D's body entered B's space and the flair can't go further than 5 feet so D is unaffected; C is also unaffected of course.
On the other hand, C's spiritual weapon doesn't really exist as anything other than a spectral image that indicates the squares it can target. It's just there as a visible reference point. If it wasn't limited in how far it can be moved in a turn, it would be considered a ranged spell attack.
If the spiritual weapon is attacking B, it can be in any square adjacent to B, or even in B's square. If the spiritual weapon is in A or B's square when it hits, there's no path for the fire shield to follow, no part of C's arm or anything C is physically attached to encroaches on B's space. If the spiritual weapon is in A's square, it's certainly not going to flair out and harm A.
So my logic is fairly simple. For all practical purposes it really is a ranged spell attack, it's a melee spell attack only because as an ongoing spell that can only be moved a limited distance on a turn. Second, no part of C or anything C is holding is encroaching on B's square, there's nothing for the fire shield to directly damage or "follow" with a flair to follow. If fire shield just flared into the square in the direction of the attacker, when D attacks B then C should have taken damage. That doesn't happen because C wasn't the one attacking.
So that's how I rule it and why. I want magic to at least make visual sense, fire shield is not "intelligent" and I think this is just one of those cases that the rules don't cover so it's always going to be a DM's ruling. At the point that B is attacked by the spiritual weapon it's no different than if A had cast magic missile from an adjacent square and should have the same ruling.
Now I have to stop procrastinating and go wash dishes. BTW, having your dishwasher crap out the morning you're having 20 people over is not good timing.