It is permitted, as long as no one can hear her.
Normally, women are not supposed to make Adhan, it is considered Mukruh (disliked) in the Hanafi Madhab. However, this ruling is due to the fact that her beautified voice is considered Fitnah. So if such Fitnah can be avoided then the ruling may change.
I have not seen this Mas’alah explicitly mentioned in the books of Hanafi Fiqh. However, some contemporary scholars say that if she is going to do Adhan for the baby in a very low voice, such that nobody is going to hear her, then the Fitnah is not present and therefore the dislike is lifted.
ذَكَرَ الْإِمَامُ أَبُو الْعَبَّاسِ الْقُرْطُبِيُّ فِي كِتَابِهِ فِي السَّمَاعِ: وَلَا يَظُنُّ مَنْ لَا فِطْنَةَ عِنْدَهُ أَنَّا إذَا قُلْنَا صَوْتُ الْمَرْأَةِ عَوْرَةٌ أَنَّا نُرِيدُ بِذَلِكَ كَلَامَهَا، لِأَنَّ ذَلِكَ لَيْسَ بِصَحِيحٍ، فَإِذًا نُجِيزُ الْكَلَامَ مَعَ النِّسَاءِ لِلْأَجَانِبِ وَمُحَاوَرَتِهِنَّ عِنْدَ الْحَاجَةِ إلَى ذَلِكَ، وَلَا نُجِيزُ لَهُنَّ رَفْعَ أَصْوَاتِهِنَّ وَلَا تَمْطِيطَهَا وَلَا تَلْيِينَهَا وَتَقْطِيعَهَا لِمَا فِي ذَلِكَ مِنْ اسْتِمَالَةِ الرِّجَالِ إلَيْهِنَّ وَتَحْرِيكِ الشَّهَوَاتِ مِنْهُمْ، وَمِنْ هَذَا لَمْ يَجُزْ أَنْ تُؤَذِّنَ الْمَرْأَةُ
Imam Abu al-Abbas al- Qurtubi mentioned in his book al-Samaa: When we say the voice of a Woman is Awrah, then people of little understanding should not think that we intend her speech, because that is incorrect. Rather, we permit non-relatives to speak to and conversate with women when there is a need for it. However, it is not permitted for them to raise, stretch, soften or cut in and out their voices because that leads to men becoming inclined towards them and stirs desire in them. Due to these reasons, it is not permitted for women to call the Adhan. (Imam Ibn Abideen, Radd al-Muhtaar).
The quote above makes it clear that the issue with women making Adhan is Fitnah. That is why a number of contemporary scholars felt that it should be permitted for a woman to make Adhan in these circumstances, since she is going to do it in a very low voice and nobody is going to hear her.
And Allah Most High knows best.
– Answered by Shaykh Noorud-deen Rashid (29.01.2021)
Is Aqiqah Sunnah in the Hanafi Madhab?