The 'Character Exchange Ticket' form the Xmas Account gift cannot be exchanged: there is no option for it in Miyamoto Musashi's Item exchange (as it is specified on the ticket).
I also checked the other Miyamoto Musashi near the Zunda booths, and the other 2 exchange NPCs near the fountain (just in case the wrong exchange NPC was mentioned), none has an option for this.
I would have uploaded screenshots, but trying to 'Upload attachment' gives an error message 'Sorry, the board attachment quota has been reached.'.

It should show this menu when you click on Musashi NPC near the Points Daruma. Did you participate in the previous years' one?



Also, as mentioned in the update, we can't pick the same partner scroll we've taken previously.

For uploading images, you can use image sharing sites like imgur.com and share by posting the bbcode link.

Only found this 3rd instance of Miyamoto Musashi later, I was coming to update my post with that.
I only started playing around the middle of the Valentine event last year, way too late to participate to it; I had seen the char appear at this location once (giveway 1 per char that time), but had totally forgotten it.

3 instances of the same NPC in the same town at the same time, with 2 of them having item exchanges, is rather messy design...

As for images hosting sites, I have used 2 in the past, both behaved very badly (for instance frequently changing link formats and/or spawning ads), then the first one closed down without warning, and later the second started to require payments; never again...