Looked for that info in Fuller's Index and learned some new stuff! No Harrah's bingo in Fallon, but here's what I did find:

Harrah's Bingo 242 N.Virginia St. 1944 with no note of games permitted.
Harrah's Club Bingo 124 N. Center St 1937 bingo only.
Harrah's Plaza Tango 14 E. Commercial Row, 1938, slots and bingo.
Harrah's Reno Club 232 N. Virginia St. 1945, slots and bingo.
Also bingo on license for Harrah's Tahoe.

Now looking at Fallon for bingo we find:
Eddie's Arcade, 1952,slots, table games and bingo on license.
Jerry's Pastime Club, same location as Eddie's in 1965, bingo
Keystone Club, 1948, bingo

Perhaps a Harrah's enthusiast(or old bingo hand...) could tell us if Bill was running an itinerat game around Reno in addition to his first humble clubs. Renting space in Fallon venues perhaps? Kinda interesting to speculate how old your card may be Scott, as I'd imagine Harrah was finding serious success in the early 50's and would seem less interested in hustling small town bingo games.