Well, I found another digital alarm clock that weighs 1.2 ounces. It has a hinged plastic cover to minimize breakage to the display, and I could probably cut that off to save 0.2 ounces or so.
This one does not look very promising. The brand name is "Digital Alarm Clock." I purchased it for $1.50 at the Daiso store. It uses a type LR44 battery, so it probably doesn't have much power for a loud alarm sound. Plus (I love this part) the instruction sheet is in Japanese English (mixed English, Kanji, Katakana, and Hiragana). There are only two button controls, so that means that the control functions are complicated.
I have several 2-ounce alarm clocks now, and I have a digital alarm wristwatch.
Too bad the old one broke.