Praise be to Allah.
Yes, it is permissible to sell animals live by weight.
The Standing Committee was asked: is it permissible to buy chickens and sheep by weight, whether they are alive or have been slaughtered?
They replied: the basic principle is that transactions between Muslims are permissible except those which are forbidden by a shar’i text. Hence we know that it is permissible to buy chickens and sheep by weight, and we know of no reason in sharee’ah why that should not be allowed.
Fataawa al-Lajnah al-Daa’imah, 13/290
And it also says in Fataawa al-Lajnah al-Daa’imah, 13/290:
It is permissible to sell sheep and other animals live by weight, whether they are weighed by kilograms or otherwise, because the purpose is to decide how much they are worth, which is decided by weight.
And it also says (13/289): It is permissible to sell animals by weight, and it is permissible to sell them by sight without weighing them, according to scholarly consensus. The permissibility of selling them by weight is not affected by what is in its stomach of food etc., because that is part of it. So it is permissible to sell them by weight.
And Allaah knows best.