Praise be to Allah.
There is nothing wrong with wearing a sling to carry a baby whilst in ihram, because that is not the kind of clothing which is described in the texts as being forbidden when in ihram, nor is it like such clothing.
It is more like carrying a water-skin or purse , or carrying one’s luggage on one's back tied with a rope around the chest. These things are not forbidden, as we shall see below.
The clothes which are forbidden when in ihram are : shirts (or thobes), pants, the burnous (a wide garment with an attached hood to cover the head), turbans, and khufuf (leather slippers).
This is indicated by the report which was narrated by al-Bukhari (5805) and Muslim (1177) from ‘Abd-Allah ibn ‘Umar (may Allah be pleased with him) who said: A man stood up and said: “O Messenger of Allah, what do you command us to wear when we enter ihram ?” He said: “Do not wear a shirt, pants, a turban, a burnous or khufuf, unless a man does not have any sandals, in which case he may wear khufuf that come below the ankle. And do not wear any garment that has been dyed with saffron or wars (safflower).”
That includes anything that is similar in nature, such as jubbahs and abayahs (kinds of cloak), short pants, hats and socks, and any kinds of regular clothing that is fitted to the body or part of the body.
Shaykh Ibn Baz (may Allah have mercy on him) said, describing the kind of clothing that the person in ihram is not allowed to wear:
“It is clear from the hadith mentioned that what is meant by sewn clothing is that which is stitched or woven to fit the entire body, such as thobes, or the upper body, such as a sweater, or the lower body, such as pants. It also includes things that are stitched or woven to fit the hand , such as gloves, or the foot, such as khufuf.” (Majmu’ Fatawa al-Shaykh Ibn Baz, 17/118)
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said:
“If a person carries a sword or a gun, that is permissible, because it is not included in that which the Messenger (peace and blessings of Allah be upon him) stated was forbidden. If he wears a belt around his stomach, that is also permissible, and if he hangs a water bottle or purse on his belt, that is permissible. The point is that the Messenger (peace and blessings of Allah be upon him) listed what is forbidden, and whatever is similar in nature comes under the same heading; whatever is not similar in nature does not come under that heading. If we are not sure about something, then the general principle is that it is permitted.” (Al-Sharh al-Mumti’, 7/152)
The sling mentioned is like what the Shaykh mentioned about carrying a water skin or carrying one’s luggage on one's back tied with a rope around the chest if necessary.
Some scholars stated that it is permissible for a person in ihram to carry his luggage on his back and to tie it with a rope around the chest if necessary. This is very much like a sling for carrying a child. (See Minah al-Jalil Sharh Mukhtasar al-Khalil, 2/308)
And Allah knows best.