How to handle the extra 12451 elements in Sophie-Germain Counter Mode?
Should the value be equal to $2^{128}$ or larger and hence require more than 16 bytes of storage, the result should be truncated $\mod 2^{128}$  


Note that this talks about the final tag value $Y_n$.  Truncating the final value will reduce security, but only slightly (e.g. possibly doubling the success probability of a forgery attempt, which is tiny even with a doubling).

Truncating intermediate values could potentially greatly increase the potential for forgeries.  At the very least, if you modify the proof giving a bound on the forgery probability, that bound will turn out to be 1 of a long (e.g. 2kbyte or longer) message.  I'm not certain how one could construct such a forgery, but the entire point of (S)GCM is that there is provably no high probability method.


  Is supporting multiplying by a 129-bit number required to handle this case?


Yup.

As an alternative, you might want to look up the Carter Wegman Counter mode; that is similar to SGCM, but uses the field $GF(2^{127}-1)$ (and divides the message into 15 byte blocks for the authentication pass).