We hear about this fear a lot but this is not how people get hiv. The way adults get HIV is very specific and involves having unprotected vaginal or anal sex or sharing IV drug needles. Any blood from a cut is not going to get into the bottle and that is an irrational fear. If you have a health anxiety issue where you frequently worry about this type of thing, it would be best to talk to a professional about it.
Your situation involves personal contact with an object in air (you feel there is a possibility that her blood ended up in the antibiotic which is still not a risk for hiv.) No worries, because you can't get hiv from personal contact except unprotected penetrating vaginal or anal, neither of which you did and you didn't share hollow needles to inject with which is the only other way to acquire hiv. Analysis of large numbers of infected people over the 40 years of hiv history has proven that people don't get hiv in the way you are worried is a risk.
HIV is a fragile virus in air or saliva and is effectively instantly dead in either air or saliva so the worst that could happen is dead virus rubbed you, and obviously anything which is dead cannot live again so you are good. Blood and cuts would not be relevant in your situation since the hiv has become effectively dead, so you don't have to worry about them to be sure that you are safe.
There is no reason for a person to test when they are safe. I doubt she dripped blood in your antibiotics anyway, but even if she did the above explains why it is zero risk for hiv even if she did it instantly.