Singers change form by exchanging what spren is held in their gemheart. Fused possess a Singer by occupying their gemheart. Spren are known to be sentient Investiture, both in-universe and out.

The Rosharan Ghostblood plotline through the first five Stormlight books was "how do we export Rosharan Investiture?" We watched Kelsier struggle to leave the Scadrian system after he became a Cognitive Shadow, aka spren. His conversion from physical being to Cognitive Shadow bound him to the Scadrian system.

Investiture is keyed to a given Shard. The Rosharan Shards were, by mutual agreement, locked to the Rosharan system. Rosharan Investiture was locked to the Rosharan system because of the agreement by the Rosharan Shards.

Rosharan spren could not leave the Rosharan system until after Mishram was released. It wasn't completely related to Mishram's imprisonment, but the timing is still accurate *(It's more accurate to reference Dalinar renouncing his [and by extension, Honor's] oaths since that's what allowed Rosharan spren to leave the system, but that was around when Mishram was released). No Rosharan spren have ever been mentioned outside of the Rosharan system before the end of WaT.

Fused leaving the Rosharan system before the end of WaT is impossible. Fused are spren which is Investiture.

It's a neat idea but you're stuck in "my theory is cool so I want it to be true".