As I was told, to get Aeronaut, you had to ride all 4 balloons to Hyde Park, Chelsea Court, The Ironworks, and Kensington Park.
I did this, and I was given the Aeronaut badge. I selected it, then later put on another badge.
When I went back to see if I could put Aeronaut back on, I could not find it.
I returned to the Ticketmaster in Digmoore Station. There was no ticket available to Kensington Park, as I was trying to re-obtain the badge. Is this a bug?
P.S. - I never finished Kensington Park, only rode the Balloon there.
My Aeronaut badge is gone too. I earned it months ago. My new character earned it and still has it. Previous characters (before the housing update, I think) have lost it.
Riding all four balloon cars does NOT replace again. I have found no way to get it back.
Note that completing Kensington has never been a requirment for it.
I HAVE completed Kensington Park and rode all four balloon rides. I used to have the Aeronaut badge enabled, and like you, as soon as I equipped another badge it disappeared out of my badge album.
I am guessing this is a bug since no one has responded to any of my former posts about why this is happening.