The Halacha has already established the principle of "Shvut D’Shvut"-when the Rabbinic prohibition of instructing a non-Jew is coupled with a requested action from him that is prohibited only by Rabbinic law, it is permitted for a Misva, major financial loss or suffering.
The question is whether this leniency applies to any case in which there is a "double D’rabanan," or only when the first Rabbinic prohibition is instructing a non-Jew. That is, is a Jew permitted to perform by himself a Melacha that is "diluted" by two Rabbinic prohibitions, in cases of Misva, loss and suffering?
Hacham David, in his Halacha Berura (p. 223), writes at length to prove that there is such a concept of "Shvut D’Shvut" by a Jew. One of his proofs is from the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) in Siman 316:19, who discusses the prohibition of "Sad"-trapping on Shabbat. If a bunch of chickens were running loose, causing damage to one’s house, he permits taking them and locking them in their cage. The basis of his leniency is that such an act is mitigated by two Rabbinic prohibitions. First, putting them in their cage is not being done for its own sake, rather to remove the damage. Hence, this type of trapping is a "Melacha She’enah Sriha L’gufa" and is only prohibited M’drabanan. Second, any trapping of domesticated animals, like chickens, is only prohibited M’drabanan. Thus, he permits a Jew to do a "Shvut D’Shvut" in a case of financial loss.
He brings another proof from the Be’ur Halacha (Rav Yisrael Meir Kagan of Radin, 1839-1933), in Siman 349. There he permits a person to carry an object in a "Karmelit" (Rabbinically designated public domain), stopping in increments of less than four Amot (cubits). This is also a "double D’rabanan." Carrying less than four Amot is only Rabinically prohibited, and any carrying in a "Karmelit," by definition, is also only D’rabanan.
He also brings a proof from Maran in Siman 325:2 that the concept of "Shvut D’Shvut" applies also to actions performed by Jews.
R. Moshe Feinstein (Russia-New York, 1895-1986), in his Iggerot Moshe (OC 4:91) applies this to a practical application. He permits picking up and carrying a child who refuses to walk on his own in a "Karmelit." Carrying anything in a "Karmelit" is only Rabbinically prohibited, and carrying a child is always more lenient, prohibited rabbinically, because of the principle of "Hai Noseh Et Asmo"- (Living beings carry their own weight). While it may not be for the sake of a Misva, it certainly is considered a case of suffering. Hacham David concurs with this ruling, in cases of a pressing need where there is no alternative.
Of course, this leniency assumes that most of our streets are only "Karmelit "and not bona fide public domains by Torah law. This is certainly true according to Hacham Bension who rules that in cases of Rabbinic prohibitions, one may rely on those lenient opinions that our streets are not "Reshut HaRabim" (public domains) by Torah law.
SUMMARY
It is permitted to carry a child who can no longer walk in a Rabbinically designated public domain.
