Doubt this would ever be possible with the current software.
Also kind of odd that you don't ever want to know your balance
I think you misunderstand. We already have the option to hide our balance on the lobby, but when you register for a game it shows your balance anyway so it kind of becomes a pointless feature. Keep in mind this is an optional feature.
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Also kind of odd that you don't ever want to know your balance
Some pros recommend checking it only monthly.