The valid inference about what was known about Bellevue's reputation based on the excerpt is:
Bellevue's mistreatment of patients was known at the time.
This inference is supported by the context of the excerpt, where it mentions that Bly was labeled insane and sent to Bellevue, and the comments from the ambulance driver who referred to Blackwell's Island as "an insane place" where "you'll never get out of," suggesting a negative perception of the facilities and treatment of patients.