In my racing days, I took a different approach. I did nothing to keep sweat out of my eyes, but I did remove the salt from my sweat. It didn't sting.
The solution was very simple (but very un-American). I kept sodium out of my diet. Consumed perhaps 1000mg/day. But this meant I ate virtually no prepared foods. 3/4s of the supermarket was simply off limits.
Prior to starting this routine, I read that the human body adjusts its salt output (via sweat and urine) to the average amount of sodium it consumes. In other words, if you regularly consume 2000mg of sodium/day, your body puts a percentage of salt your urine and sweat to add up to 2000mg at the end of the day. But our bodies only have a very broad based "sodium-a-stat". It cannot react to what you are doing now. So a person who consumes a lot of sodium will be placing a large percentage of sodium into his/her sweat. The person who stays near sodium free will have near salt free sweat. No salt in the eyes. Also very little need for salt intake on very hot days. (Potassium doesn't play by the same rules. It must be replaced by everybody.)
That this works can be seen simply by looking at other parts of the world. There are people who live in desert climates with no local salt sources. Salt may be very expensive, an item for the wealthy. The poor? They don't eat it and don't need it. We westerners go there and if were aren't hanging with the wealthy, we'd better bring our salt or we suffer. And in contrast, my dad's stories of marching through Texas in the Army, WW2. Very high salt diet. More was consumed before the hot marches. The issued blouses streaked white down the back from salt. Those who consumed too little keeled over. Got rounded up by the medics, given salt intravenously (very, very unpleasant according to my dad) and sent back out.
Lastly, the low salt routine is NOT a quick fix. It takes many months to retrain the body to not excrete sodium. You have to taper down gradually or riskj sodiun deficit. (I found that came fairly naturally. It took my months to identify all the sources of salt; it is so pervasive.) I warned that this was un-American. But there is a real plus. Your jersey will stay far more pleasant!