Mostly (as in very very high percentage) no.
In a very low chance scenario, yes (and TL;DR: it's as Delta Blast Burn says)
[I understand you (OP) may understand how this works; do note I'm also breaking this down for other readers]
Explainer (wall of text and semi-technical jargon alert):
I don't know the specifics of how, but I know it does in fact change the PID as necessary to retain non-shinyness. I'm pretty sure, as you experienced, it only changes a single bit.
As I expected, caused by changing mystatus.
It falls in-line with a previous report.
Consider this hex edit to your mystatus block, then import it using PKHeX.
(from: https://discord.com/channels/343093766477053953/367479180398428170/823116547970826260)
Lemme know if it works or doesn't work.
I had in mind to send my Shiny Mew to my HOME account too, but I forgot to log into Pokemon GO and I have now to restart the 30 day count. I guess my HOME account is going to wait a bit more.