Will a bladder biopsy help my doctor figure out if my bladder infection is very serious?

Its Contraindicated. Unless a cystoscopy must be perfomed emergently, the procedure is contraindicated in the presence of a bladder infection. A cystoscopy and bladder biopsy could lead to a massive infection, sepsis, and possibly death if performed while there is an untreated urinary infection.
Maybe. Bladder biopsies help identify malignancy in the bladder. If there is concern for infection, then a urine culture is needed to identify the type of infection and medicaiotns that will be effective in trating the infection. A biopsy by itself will not generally tell you about an routing bladder infection.