Do you routinely recommend diagnostic endoscopy for patients with persistent enterococcus bacteremia despite receiving adequate antimicrobial therapy and no clear nidus?

It depends. Did you do an echocardiogram to rule out endocarditis? Urine cultures were negative? Gallbladder ultrasound was negative? CT of the abdomen and pelvis with contrast was negative?
Any other symptomatology that accompanied the recurrent episodes of enterococcus bacteremia that could help us...

Probably should more often but don't. The reason is the affected patients are often too fragile for an endoscopy and may be at high risk from the procedure. Given newer technologies this may be more feasible.

Yes, looking for an underlying malignancy.