Sorry to badger you but could I have an answer as it’s been over a month.
My apoligies that it took so long to get back to you.
Regarding your question, it is permitted for another person to pick up the baby. The reason is because there is a rule regarding the halacha of benefiting from a melacha that was done by a Jew on shabbos, called “maaseh shabbos”. This rule only applies to actions that are clearly forbidden, but not to actions that are controversial if they are forbidden or not. The reason for this is because we apply the rule of safk d’rabonon l’kulah, and since the rule of maaseh shabbos is only d’rabonon, we will rule leniently and say that it is permitted. There is controversy if maaseh shabbos applies to a melacha that was done, but it didn’t affect any real change to the item, such as when mileches hotzah was done. Although a forbidden action was done, however the item that was carried, didn’t change at all, therefore according to some rishonim the prohibition will not apply. Even though there are other that say that it does apply, however still it is controversial and since maaseh shabbos is d’rabonon we are lenient and it is permitted to use the item that was carried. Therefore there are poskim that rule that maaseh shabbos does not apply to meleches hotzaah. (This idea is controversial in itself, as there are poskim that are stringent and say that it is only permitted to use if the bringing the item only transgressed an issur d’rabonon, and specifically if it was accidental.) However in our case though may be other permitting factors.
If the person who brought the pram thought that there is an eruv, even if he is going according to opinions that we don’t agree with, nevertheless the action of bringing the pram would be a shogeg, because he thought that action was permitted on shabbos, coupled with the fact that most streets nowadays are only a carmilus, which is not a reshus horabim d’orayso, therefore it is a shogeg by a drabonon, and permitted according to everyone. Secondly, if the child is able to walk on his own, we would have the rule that “chai noshe es atzmo”, and taking him in the pram would only an issur drabonon, even if the pram was taken through a reshus horabim.
There are poskim that say that the idea of maaseh shabbos does not forbid one to benefit from a person. Meaning that if a pot of food was cooked on shabbos we cannot benefit from that food, however if a person was brought here on shabbos, even if it was in a forbidden way, (he drove here) it is not forbidden to benefit from the person. (Moehr Hashabbos chap. 18 ftnt. 53). This would apply especially here that one person cannot make another person forbidden, no worse than the rule that one person cannot make another person’s object be forbidden to him without his consent or knowledge, (See Imrei Yosher 1-129, Maharshag 2-130).
Aside from all of this, I am skeptical if holding a child is considered benefiting from the child. When someone holds a child, the child is not providing him with a service that we should say that he is benefiting from him, rather the adult is providing the child with the service, and because you like the child you are enjoying your giving to the child.
Rashba Shabbos 130b in the name of Rabbeinu Yona, Rambam Hilchos Shabbos 6-24, Rav Y. Cahen shlit”a, See Chayei Adam 9-11 brought in Biur Halacha 318 D:H Achas,