What antibiotic/treatment is best for pseudomonas bacterial sinus infection?

The only way to know for sure what bacteria is causing your sinus infection is by taking a culture of your nose.  I assume you haven't had this done because if you really had pseudomonas you wouldn't be put on bactrim (doesn't cover pseudomonas). Bactrim is usually used to cover MRSA.  Is your ENT directing this treatment?  Sometimes antibiotic compounded sinus rinse can help. I also sometimes have patients use a few drops of baby shampoo in their sinus rinse to help break up biofilm.  Make sure you talk to your ENT before doing any of this.