Archaic phrase similar to "not give a damn"

Here are some options (with examples near the 1930s):

• Don't care an atom (1847, 1909)
• Don't care a button (1919)
• Don't give/care a doggone (1923, 1939)
• Don't care a (red) cent (1919, 1924)
• Don't give/care a curse/cuss (1923, 1936)
• Don't care a (brass) farthing (1900, 1922)
• Don't give/care a hang (1929, 1933, 1939)
• Don't give/care a hoot (1921, 1939)
• Don't give/care a fig (1914, 1947, 1948)
• Don't care an iota (1918, 1947)
• Don't care a jot (1922, 1944)
• Don't give a mugger (1945)
• Don't care a peascod (1934)
• Don't care a penny (1920)
• Don't care a pin (1925)
• Don't care a rat's tail (1901)
• Don't give/care a rap (1901, 1933)
• Don't give/care a rip (1922, 1924)
• Don't care a straw (1912)
• Don't care a tittle (1899)
• Don't give/care a toss (1925)
• Don't give/care a traneen (1898, 1988)
• Don't care a whit (1923, 1929, 1935)
• Don't give/care a whoop (1930, 1940)

Further alternate forms of this are "don't care one X" and "don't care two Xs". I also found one example that mixes things up and "doesn't give a red hoot" (1923).

As you can see, there are essentially two identically-worded snowclones here: one used with "expressions of anger", euphemistic or not (bugger, doggone, curse, cuss, (god)damn, fuck, hang, hoot, mugger, whoop) and one that's used with worthless things ((red) cent, fig, iota, jot, penny, pin, rat's ass/tail, rip, straw, toss, whit). If you follow this format you could even create your own.

You can probably use “don’t give a damn”, which was already in place in the ‘30s.

(don’t) give a damn:

• The above phrase was originally I don’t give a dam (yes, the n is missing on purpose) and seems to have been brought back to England by military men traveling to India in the mid 18th century. A dam was an Indian coin of little value. After spreading to civilian usage, the phrase changed to. I don’t give a damn and was first recorded in America in the 1890’s. (word-ancestry.livejournal.com)

A less strong and informal expression which was already used at that time is:

I don’t care:

From The Ladies' Home Journal - Volume 43, Issue 1 - Page 158 - 1926 - ‎

I swell with pride for my boys when I consider the girls they have married. But I do not make demands. I do not suggest that my son's wife call me "mother," unless she wants to. I do not care what she calls me. I do not ask that she say she loves ...