From Eric Partridge, A Dictionary of Slang and Unconventional English, fifth edition (1961):
apple-sauce. Impudence: mostly lower middle class: late C.19–20. An elaboration of sauce, n., 1 ...
The relevant definition of sauce in Partridge is as follows:
sauce. Impudence, impertinence: coll[oquial] and dial[ectal]: 1835, Marryat (OED); perhaps much earlier (see [more] sauce than pig [defined in its own entry as "(To be) very impudent, impertinent: coll[oquial]: late C 17–18. B[ritish] E[nglish]"]. Ex saucy ...
J.E. Lighter, The Random House Historical Dictionary of American Slang (1994) offers this more elaborate, two-definition entry:
applesauce n, 1. Theat[rical] silly trite comedy [Citations, starting in 1918, omitted.] 2. nonsense, flattery, insincerity; lies—also used as interj[ection]. [First two citations:] 1919 T.A. Dorgan, in Zwilling TAD Lexicon 15: They spill a lot of applesauce about big money, 1920 in Collier's (Jan 1, 1921) 18: That's all apple sauce!
So applesauce, as used at the end of the OP's quotation simply means "baloney!"
A Google Books search doesn't find the Dorgan/Zwilling example mentioned in Lighter, but it does find two instances (both by the same author) from Collier's Magazine in 1921. From H.C. Witwer, "Auto Intoxication," in Collier's (January 1, 1921):
"Gimme my money back, you burglar!" shrieked the newcomer, ignoring me entirely. "I'm gonna have you pinched. This old tomato can is nothin' but a mess of junk! When I cranked it up last Friday mornin' the rear end fell out in the street, and the repair man tells me the only reason it ever run a foot is because you went to work and doped the motor with ether! The bearin's is all shot, it needs—!"
"Ssh!" interrupted the dealer wearily, as one who hears an old story. "That's all apple sauce! I told you the job needed the touch of a monkey wrench here and there. You can't expect to git no factory pet for what you paid for this car. They's no use gittin' hysterical, you bought the car as is, and they i absolutely no comeback. If you don't want it, leave it here and I'll sell it for you some time to-day and git you what I can for it!"
And from H.C. Witwer, "The Shooting Stars," in Collier's (June 11, 1921):
"A thousand pardons, old man!" says my vis-a-vis, climbin' out into the road. Dug, he didn't look so particularly hugely when he was scrunched up behind the steerin' wheel, but when he stood up and unfolded his full length, it was different. "You were hitting it up so fast, I had no idea you intended turning off the road here," he goes on. "However, I'll—"
""That's all apple sauce!" I bellers, steppin' over to him, "A guy drivin' a can like yours should be prepared for anything, and they ain't nobody in the world goin' to run me down and get away with it. Put up your hands, you big stiff!"
These are the earliest two matches in Google Books search results, although the expression is quite common by the mid-1920s, it's euphemistic qualities making it a favorite choice in periodicals such as Boys' Life.