Green’s Dictionary of Slang

bat n.5

[batty adj.1 /bats adj., although it could be a result of a bat n.3 (3)]

in pl., insanity, esp. manifested in a drinker’s delirium tremens.

[US]Number 1500 Life in Sing Sing 246: Bats. Delirium tremens.
[US]G. Henderson Keys to Crookdom 397: Bats. Crazy, delirium tremens.
[US]‘Iceberg Slim’ Pimp 156: A stud would have to be slick as grease to plant bats in the skull of a bitch that was sane.

In compounds

bat house (n.)

(US tramp) a psychiatric institution.

[US] ‘Hugh McHugh’ Back to the Woods 38: I could see myself in the giggle-giggle ward in a bat house.
[US]Pacific Reporter 165 1152/1: Affidavit Tom [...] clearly indicates that he should be sent to the bug house, the crazy house, the foolish house, the bat house, the looney house, the mad house, the nutty house.
[US]A.J. Barr Let Tomorrow Come 267: An insane asylum [...] bathouse.
[US]Irwin Amer. Tramp and Und. Sl. 22: Bat house – An insane asylum ; a place where are confined those ‘batty’ or with ‘bats in the belfry,’ which expression has been shortened in vagabondia to ‘bats’ or ‘batty’.
[US]Ragen & Finston World’s Toughest Prison 790: bat house – Insane asylum.
[US]T. Swerdlow Straight Dope [ebook] Did she tell you her mother was in the bat house?

In phrases

off one’s bat

out of control.

[US]R.E. Howard ‘Fist and Fang’ Fight Stories May 🌐 ‘All right, you black swine!’ I yelled at him, kind of off my bat, I reckon. ‘Go ahead—do your worst!’.