Would Flucloxacillin antibiotic make my Botox wear off quicker as I am hesitant to take it?

This antibiotic should not affect the Botox duration. If your physician is recommending you take an antibiotic for an infection, I would listen to them.

Antibiotics should not effect the longevity of your Botox!  Don't worry if you need to take them.  On the other hand, if you are due for Botox and are very sick, I would delay your treatment until you are feeling better.