Jelly, 16 bytes

J:g¥’}U×ḷb§¥ƬṪS%

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A less naive computation of the digital roots comes out a bit longer by having to special-case 0. Inverted output: truthy (nonzero) if the checksum fails.

J                   1-indices from the start
 :                  divided by
  g¥                their GCDs with
    ’}              B - 1
      U             reversed
       ×ḷ           times each N.
            ƬṪ      Loop while unique:
         b ¥        Convert each to base B
          §         and sum each.
              S     Sum
               %    mod B.