This could be plausible. The USSR was way behind on subs, even with the Toshiba/Konigsberg scandal. However, this race is one with more immediate implications than the moon. Both the Soviets and the Americans would be greatly interested in undersea mineral deposits. Deep-diving subs would make great weapons platforms...and espionage platforms. Plus, add in the driving Hyman G. Rickover against the driving Sergey Gorshkov...and you make it personal!
And, what would victory consist of? Would the Trieste's dive win it?