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That would be a possibility. But not all variants that do allow turn pass associate this turn pass with a piece. And when different pieces can do a null move, you would have different notations for the same move. Of course you coould always associate a null move with the King, which should always be there.

But the conventional solution used by interfaces for orthodox Chess is to write them as 'pass' or '--'. Although in Chess these are of course illegal, many interfaces allow them during interactive analysis. Where it can be important to answer the question "what is the threat I am facing in this position".

Th problem here is not how to write the turn pass, but that the I.D. currently is not aware at all that when two moves of the same player are done in a row a turn pass should be inserted between them. Hence the notation gets 'out of phase'. Which is fatal for reading it back, as SAN does not explicitly encode the color of the moving piece. Nf6 could mean either a white or black Knight. For two Knights of the same color being able to make that move it would add disambiguation, like Ngf6, but not if a Knight of the opponent can make the same move. When it encounters Nf6, it only looks for Knights of the color that it expects to be on move, which only follows from the sequence number of the move in the game record to be even or odd.

So I guess that before adding a move to the game it should first check whether the color of the moved piece corresponds to the even/oddness of its sequence number, and if not insert the turn pass before adding the move.