Is it permissible to use a cleaning product labeled "alcohol-free" with 0% alcohol content, but which contains ingredients chemically classified as alcohol, such as linalool, and to pray while using it, knowing that it is an important hygiene product and no alternative has been found?

We have no knowledge of the nature of these materials. If you suspect that the product contains alcohol, which is ruled to be impure, the default is that it is pure. It is not ruled impure merely based on suspicion. Therefore, it is permissible to use it until you are certain that it contains something that prevents its use.