Jelly, 42 bytes

l2l2Ḟ
WµÇ2*2*⁸d⁹jɗ×nJaƲ¹Ƈ)F$ÐLÇ»-ị“4HBSI12

A full program that accepts a positive integer and prints the spoken binary.

Try it online! Or see the test-suite.

How?

l2l2Ḟ - Link 1: integer, X -> floor(X log 2 log 2)
          Note when X = 1 l2l2 -> (-inf + nan j) and Ḟ takes the real part -> -inf

WµÇ2*2*⁸d⁹jɗ×nJaƲ¹Ƈ)F$ÐLÇ»-ị“4HBSI12 - Main Link: positive integer P
W                                    - wrap P in a list
                      ÐL             - apply while distinct:
                     $               -   last two links as a monad - f(Current):
 µ                 )                 -     for each {V in Current}:
  Ç                                  -       call Link 1 -> floor(V log 2 log 2)
   2*                                -       2 exponentiate {that}
     2*                              -       2 exponentiate {that}
                                               -> value of the digit to use, say D
       ⁸    ɗ                        -       last three links as a dyad - f(V, D):
         ⁹                           -         D
        d                            -         {V} div-mod {D}
          j                          -         join {that} with {D}
                Ʋ                    -       last four links as a monad - f(Parts=that):
              J                      -         indices {Parts} -> [1,2,3]
             n                       -         {Parts} not equal {that} (vectorises)
            ×                        -         {Parts} multiply {that} (vectorises)
               a                     -         {that} logical AND {Parts}
                 ¹Ƈ                  -       keep truthy values
                    F                -     flatten
                        Ç            - call Link 1 (vectorises)
                         »-          - max with -1 (convert -inf to -1)
                           ị“4HBSI12 - index into (1-based & modular) "4HBSI12"
                                     - implicit print