I had never thought of trying this! I brought up my FC (Brynhildr "FFXIV Pals") sure enough. The good news is is FC shows up! The bad news is my Mister Jingles doesn't even show up on the list of members....
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Try a character rank change in the FC, it maybe can force update the lodestone. It can take a few hours to display but it worth tryin'
It probably doesn't matter if you don't want to pay for it, but you could probably fix it by transferring back to Phoenix and then transferring back to Brynhildr again.
I changed my characters name to the following "Mr Jingles@Brynhildr" but the name change does not seem to have worked. I still can't find him on Lodestone and "Mr Jingles" on my Mogstation just links to the Mister Jingles@Phoenix page still. I will give it a few days to see if it just takes awhile for the change to take place but it isn't looking promising.
I think the lodestone updates a lot quicker than it used to, so it should only take an hour now.
So I decided to try SE Support one more time to see if there was anything they could do to help. Here is what was SE Support said:
"Please advise that [Mr. Jingles] on [Brynhildr] is essentially [Mister Jingles] on [Phoenix]. Mister Jingles did start out on Exodus and then moved over to Phoenix on 10/02/2013 with a world transfer. Then again, from Phoenix to Brynhildr on 12/01/2014...I looked up Mister Jingles on Phoenix on our end and found that the character is completely deleted, leading this to be a possible bug on the Mog Station end...The best thing to do right now would be to submit a new bug report for this information, that Mog Station is displaying your character twice"
Basically, I am stuck hoping the engineers that manage MogStation/Lodestone will see the bug report and fix the issue. I reported the bug in both the forums and through https://sqex.to/vzc but I am not planning on holding my breath...