Before I went to the bath house, I found a Subeautique invitation randomly. I locked it in my inventory and then went to the bathhouse, and it said I won a Subeautique invitation. However, the second invitation isn't showing up in my inventory.

The subeautique invitation is automatically added to your currencies =)

OK, I checked there, but it's still only showing just the one. :

It should be one in your currencies (unless you had redeemed more) and one in your inventory.

So you don't get a physical invitation from the bathhouse. If you don't think you were credited for it, you can file a ticket and they can take a look into it for you :)

Oh. OK, wow, how confusing. XD Thank you!

No problem =)