All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
If the alcohol in this medicine can cause drunkenness if drunk in larger quantities, then using any smaller quantity of that is forbidden for mouthwash, etc. It is also forbidden to sell it, buy it, or benefit from its price. The evidence is the Hadeeth reported by At-Tirmithi that the Prophet said: “Any drink that causes drunkenness in large amount, then a small amount of it is forbidden.”
Another narration reads: “If something that fills a whole receptacle causes drunkenness, then a handful of that thing is forbidden.”
Yet, another narration reads: “Even a mouthful of it is forbidden.” [Abu Dawood]
But if the amount of the thing that causes drunkenness was used up, or dissolved, or if it had changed through manufacturing to a different substance and if its smell, taste or ability to intoxicate have disappeared, then it is not forbidden and it is lawful to sell it, buy it, use it…etc.
For more benefit on the existence of a small percentage of alcohol in food or other substances, please refer to Fataawa 382309, 327754, 299059, 301753, and 81423.
Allah knows best.