Institution without kosher food:[1]
With respect to placing such an individual in an institution for the mentally impaired that does not provide kosher food, the Poskim[2] rule that it is permitted to send a child with severe intellectual disability who is below the age of Mitzvos to a non‑kosher facility.[3] Furthermore, if there is a chance that the individual will receive a cure in the institution and return to normalcy, then certainly he is to be sent there, even if he is above the age of Mitzvos.[4] However, once the child regains normal cognitive functioning, he is to be removed immediately from any institution that provides him with non‑kosher food.[5] This leniency, however, applies only where the individual is deemed completely incompetent (shoteh gamur). However, if the individual is not fully incompetent and possesses some capacity for understanding and education in matters of Torah and mitzvot—such as individuals with Down syndrome or similar conditions, or a cheresh[6] (deaf individual) who is capable of speech and comprehension—then one should refrain from sending them to such institutions, as they are amenable to at least partial religious instruction and observance.[7] [However, from the letetr of the law it is permitted even in such a case, as explained in Halacha 8f that for medical reasons one may instruct a gentile to feed a child even Biblically non-Kosher food.[8]]
[1] Chasam Sofer O.C. 83 and in Hagahos 343:11; Maharam Shif O.C. 163; Orchos Chaim 343:1; Kaf Hachaim 343:22; Igros Moshe 2:88; Shevet Halevi 9:4; Minchas Asher 2:47; Piskeiy Teshuvos 343:10; Toras Hakatan 20:11-17
[2] See Chasam Sofer, and Maharam Shif ibid, brought in Kaf Hachaim ibid that although it is permitted to send the child to such an institution until the age of Mitzvos, it is better not to being that the non-Kosher food will contaminate his soul.
[3] The rationale is that this is considered indirect consumption of non‑kosher food rather than direct feeding, which is prohibited. [Chasam Sofer ibid]
[4] Chasam Sofer ibid; Orchos Chaim ibid; Kaf Hachaim ibid
[5] Chasam Sofer ibid; Piskeiy Teshuvos ibid footnote 130 that this is a Chumra and not from letter of law, as he is being fed by gentiles
[6] See the following Poskim regarding the status of a Cheresh today when they have educational systems which teach them to talk: Tzemach Tzedek of Nikolsberg; Shut Shaar Hazekeinim 2 Beis Melo p. 135; Orchos Chaim 343:1; Kaf Hachaim 343:23; Kesav Sofer; Mishpitei Uziel; Minchas Shlomo 1:34; Piskeiy Teshuvos 55:14
[7] Chasam Sofer ibid and ibid; Minchas Shlomo ibid; Minchas Asher ibid; Piskeiy Teshuvos ibid footnote 125-127
[8] Piskeiy Teshuvos ibid footnote 127