Seems believable as I've found a recipe for it in the 1966 Booth’s Handbook of Cocktails and Mixed Drinks.
https://www.scribd.com/document/408647965/Collectif1806-1966-BOOTH-S-HANDBOOK-OF-COCKTAILS-AND-MIXED-DRINKS-E-pdf#content=query:balalaika,pageNum:83,indexOnPage:0,bestMatch:false

Supposedly, the first appearance of the Balalaika was published in the 1965 revised edition of The Savoy Cocktail Book. I haven't been able to confirm this myself, but the Japanese cocktail book 新版 バーテンダーズマニュアル credits it.

A much better name than "Gin-Vision Bell," the gin riff on the mezcal base "Division Bell."

Both W.J. Tarling's 1937 Cafe Royal Cocktail book and Harry Craddock's 1937 UKBG Approved Cocktails list a similar cocktail, the Bermudiana Rose, which includes lemon juice. Stan Jones, in his 1977 Complete Barguide, lists a variation under the Bermuda Rose that includes "3/4 oz lemon (or lime) juice," which is quite similar to his "Flamingo" recipe.