Hi again Dantheman ! (and any others interested.)
This is an addendum to my post #4.
First, there must be some amount of iron in a metal alloy for the metal to be attracted to a magnetic fields lines of flux.
Prior to the modern technology of using plated harness rings for livestock, solid brass alloys were typically used, however, as you thought, it's entirely possible your ring is made from the Red brass high-copper alloy C23000 listed below, which figures 83.95% - 85.95% copper, the highest copper percent of the three brasses listed and maybe your magnet just wasn't quite strong enough to detect the small 0.05% of iron.
If you go to the Source Link below, you will see there are only (3) types of
Brass metal containing iron that might cause some attraction to a powerful enough magnet, the amount of attraction depending on the percentage of iron in each case.
I copied and pasted these three types of brass below for your review and this information is by a highly credible source and was just updated January 27, 2019.
SUBJECT: Brass Alloys containing iron
SOURCE: https://www.thoughtco.com/common-brass-alloys-and-their-uses-603706
Aich's alloy
60.66% copper, 36.58% zinc, 1.02% tin, and
1.74% iron. Corrosion resistance, hardness, and toughness make it useful for marine applications.
Muntz metal
60% copper, 40% zinc, and
a trace of iron, used as a lining on boats
Red brass
American term for the copper-zinc-tin alloy known as gunmetal considered both a brass and a bronze. Red brass usually contains 85% copper, 5% tin, 5% lead, and 5% zinc. Red brass may be copper alloy C23000, which is 14 to 16% zinc,
0.05% iron and lead, and the remainder copper. Red brass also may refer to ounce metal, another copper-zinc-tin alloy.
I hope you pursue obtaining a neodymium magnet, as they are the best for testing coins and metals in general. If you are still having trouble obtaining one from a discarded computer hard drive after asking family and friends if they have an old computer tower sitting around that they have been meaning to dispose of, then you could always purchase a neodymium fairly cheap at
https://www.kjmagnetics.com/.
If you get one, be sure to keep it away from any electronic equipment with data stored on magnetic media, as it can wipe it clean!
However, you can make an inexpensive storage container for a neodymium that will reduce the outflow of magnetic lines of flux by using a cast iron pipe with both ends threaded and caps screwed on the ends. Magnets have a positive pole at one end and a negative pole at the other end. The majority of magnetic lines of flux will seek the path of least resistance and travel in the pipe walls to make connection with the opposite pole of the magnet, thus reducing the amount of flux lines that escape through the walls into the outside air.
The thicker the pipe walls, the less flux lines will escape outside the container. The thickest walls I could find for my container were about 3/8" from a local lumber and hardware store, who also threaded the ends for me, but you might find thicker walled pipe at a metal recycler or junk yard.
Here are a couple of links that will give you animated visuals and text describing what I've explained above. On each website just scroll down to the sub-title indicated.
http://www.coolmagnetman.com/magshield.htm Sub-title: Containing the Field
https://www.kjmagnetics.com Sub-title: How thick should my shield be?
If you keep the container several feet away from any equipment with magnetic media, there should be no problem. Safety precautions on the use and handling of neodymium's are available on the Internet and should also be on the K&J Magnetics website hyperlink above.
Good luck!
ToddB64