Green’s Dictionary of Slang

1. to persist in one’s demand.

1884		(con. c.1840) ‘Mark Twain’ Huckleberry Finn (2001) 72: I stuck out for mo’ dan fo’ dollars, en I said ’f I didn’ git it I’d start a bank mysef.
1900		Boys Of The Empire 11 Dec. 149: I had to stick out to the doctor that you was, in order not to split and to account for the broken window.
1978		F. Norman Dead Butler Caper 125: Having early-morning tea brought to my room was the only condition I’d stuck out for when I agreed to stay in the house.

2. to argue with.

a.1854		‘Cuthbert Bede’ ‘Æger’ in College Tales (1893) 220: I must stick out that I’m somebody else.
1891		Punch 14 Feb. 84: Second S.G. [...] You can stick ’er out it wasn’t you.