Praise be to Allah.
First of all, we ask Allah to grant you success in your field, and to make obedience to Him dear to you, and make disobedience to Him hateful to you.
There is nothing wrong with you using hand sanitizers, even if you get some smell like perfume on you as a result, because the intent is to sterilize and sanitize, not to perfume yourself. Sterilization is an important medical necessity, according to medical safety protocols, so as to prevent the spread of disease and epidemics, and to protect wounds from infections and complications. It is not permissible to ignore this necessity because of something that is prohibited due to a possible outcome, and is not prohibited in and of itself.
This applies even if we assume that the prohibition is applicable in the scenario mentioned in the question, so how about when the conditions of prohibition are not applicable in that scenario?
That is because the fragrance in hand sanitizers is very mild; it is not strong and does not spread widely, and it disappears within a few moments. That fragrance is mixed with the strong and piercing smell of the sanitizer itself, so the effect of the fragrance – if it is present – grows weak and men would not pay any attention to it in most cases, especially in hospitals in which the smells of disinfectants and cleaning liquids that are used in their hallways prevail, and in which the workers are used to using hand sanitizers.
As the fuqaha’ have permitted women to use deodorants to remove unpleasant smells from their bodies, even if they have a light fragrance, because the purpose of removing unpleasant smells is greater than the purpose of avoiding possible bad consequences of using a mild fragrance that does not provoke desire and does not cause temptation, it is more appropriate to say that it is permissible for a woman to use hand sanitizer even if it does contain some fragrance, for the reasons explained above.
In a commentary on Nihaayat al-Muhtaaj (2/340), which is a Shaafa‘i book, it says:
It is recommended for a woman to remove unpleasant smells, even if what she uses has a fragrance, because there is no other way of achieving that result. End quote.
This has been discussed previously in the answer to question no. 155211.
And Allah knows best.