I found one (ugly) way to do it:

Instead of directly changing the textvariable contents, I delegate this to a slightly later point using an "after" call. So instead of the line

  $textvar = $new & 3;
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I use this code line:

  $mw->after(10, sub { $textvar = $new & 3 });
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Btw.: Doing the change of the textvariable contents in the invalidcommand callback also turns the validate option to 'none'.