These are complicated matters that I often fail to understand, so someone please correct me if I’m wrong, but…
Wouldn’t all sides agree that it would be outright heresy to say that the Holy Spirit proceeds hypostatically from the Son as well as the Father? That’s been my impression; thus, I don’t think the filioque is meant to be claiming such a thing.
There was a thread on this recently where some very detailed discussion was going on, particularly, if I recall correctly, between Cavaradossi (who is eastern Orthodox) and Mardukm (who is an Oriental Catholic).
The argument winds up conflated because of a lot of linguistic issues, but at its core, I think it has to do with causality. The Florentine decree, which I suppose would still be the official Catholic teaching on the matter, attributes the two Greek terms aitia (cause) and arche (principle) to the Son. All of the other disagreements (whether the subsistence, hypostasis, energies, etc. of the Holy Spirit come from or through one person or another) are in some sense secondary to the one over causality, because they follow the argument over causality.
Among the Greek tradition, it was almost unanimously confessed that the Father is the only cause and principle. Even Gregory of Nyssa, who is the most
Filioque friendly Cappodocian Father, in terms of proof-texts, does not ascribe these two terms to the Son. Instead, he divides the persons into the cause and the uncaused, and then of the uncaused, he distinguishes between the two by having one of the caused (the Spirit) being from the cause (Father) through the other caused (the Son), while the other caused (the Son) is from the Father through none other.
Gregory the Theologian, on the other hand (the Cappodocian Father used against the
Filioque the most), speaks of the Father as the cause, the Son as the begotten and the Spirit as the proceeding. This distinction between begetting and procession, originally made to combat the Eunomian thesis that the Son and Spirit cannot both be caused by the Father, as the Spirit would be a second Son, preserves the connection of the Spirit to the Son without having to engage in Gregory of Nyssa’s more speculative approach to differentiating the two caused persons. Instead of cataphatically attempting to define the difference like Gregory of Nyssa, Gregory the Theologian apophatically defines procession as a mode of generation which is not begetting, pointing out that we should refrain from attempting to define cataphatically procession when we cannot even do the same with the generation of the Son.
To sum up over fifteen hundred years of theological thought, to this day, the East still rejects ascribing any sort of causality to the Son, because causality and principle are hypostatic properties of the Father.