I'm wondering if there is a flight badge for the B-Wing in the original X-Wing? I've been going through the pilot proving grounds and got the badges for the X-Wing, A-Wing and Y-Wing, and I got through level 08 on the B-Wing course and it didn't award me the badge. I thought maybe it was because the first time I played it I accidentally skipped a few gates (even though that didn't impact me getting the badges on the other ships), so I re-did it and made sure not to skip any gates but I still didn't get the badge after completing level 08. Am I doing something wrong? Is there a badge? I'm playing the 1994 Special Edition of X-Wing.

Yes there is, but the B-Wing flight badges (along with B-Wing historical mission badges, and awards from the Imperial Pursuit and B-Wing campaigns) are added to your Medals Case since they ran out of room on the uniform.