It's safe to say I lucked out with this lot. Three bromides, all premium prizes. One of them is uncataloged, even!
The player above is Shigeru Chiba of the Yomiuri Giants. The top line mentions the league and team name, while the bottom line is the player's name. The backs are blank.
Engel doesn't mention any 4th Prize premiums, and based on my experience I'm guessing 4th Prize was just another regular bromide. 3rd Prize are larger - 2-15/16" by 4-1/16". The card you see above is a 2nd Prize, at 4-7/8" x 6-7/8", which makes it a good bit larger than a postcard. That would make it perfect for displaying on a kid's wall. The player above is Hiroshi Ohshita (or Oshita) of the Flyers. The text line has the player's name followed by the team nickname in parentheses.
Both my 2nd and 1st Prize cards from the 1950 Bromide Game set have tape/glue remnants at the top on the back. This could have been from the owner displaying it in a book or on a wall, but given how bromides were sold, these could have been attached to the packaging/display for the bromides.