I thought I'd chime in here. First, to answer the original question, it is impossible to choose "one" agent for neutropenic fever among the anti-Pseudomonals, given different risks in different patients (prior ESBL colonization, MDR Pseudomonas colonization, allergies, prior antibiotic exposure - just a few things that come to mind). You can certainly offer guidance on when to use one agent over another, but having only one option just isn't possible.
To some of the other comments: the increased risk with cefepime in febrile neutropenia has been debunked and was mostly associated with inadequate dosing. The FDA did a nice followup with this. I'd imagine it's the first line agent in most institutions with a sizeable cancer population and it certainly is in mine. I would also say that cefepime is just about as far as you can get from a narrow spectrum agent, and the IDSA guidelines (updated in 2011) were completely aware of the Yahav meta analysis (published in 2007) when they were released. The NCCN guidelines, which are updated more frequently, still strongly recommend cefepime as first line monotherapy.
The aminoglycosides aren't recommended not because of their toxic profile (although that does factor in) but because they have been shown to be less effective in comparison to beta lactam agents for bacteremia and have additionally been shown to be less effective in neutropenic patients (the case with all antibiotics, but probably moreso with aminoglycosides than with the beta lactams). Aminoglycosides can still be quite useful though if the patient has a history of an MDR organism or is at high risk for such an infection, but they generally aren't recommended to be used front line.
Vancomycin, or MRSA coverage in general, is not recommended first line unless specific risk factors or present (indwelling line, prior colonization with MRSA, sepsis, etc.) or if the patient presents with a condition such as skin/soft tissue infection or healthcare-associated pneumonia that otherwise warrants coverage. Even here this is probably excessive and led to significant overuse of these drugs. Levofloxacin, to which MRSA rapidly develops resistance and has high baseline resistance, is not appropriate for MRSA and isn't recommended in any guideline I know of. Additionally, many of the highest risk patients are on quinolone prophylaxis to begin with (depending on your institution's usual practice). Bactrim is fine for non-invasive infections, but a great RCT comparing Bactrim to vancomycin for such infections showed that patients treated with bactrim had overall higher mortality. So that's probably not a great choice either if concerned for a serious MRSA infection in a neutropenic patient.
Imipenem isn't a terrible option, but it's somewhat less potent against Pseudomonas than meropenem and has generally fallen out of favor due to more complicated dosing and worse adverse event profile. It's also technically not correct to say that the kidneys "clear" imipenem faster, rather, dehydropeptidase which is found in renal tubules metabolizes imipenem. Hence the addition of cilastatin.
Long story short, cefepime, pip-tazo, and anti-Pseudomonal carbapenems are all appropriate front line options for monotherapy in febrile neutropenia, provided all are dosed at their maximal dosing and risks are taken into account. MRSA coverage isn't recommended in all cases, but is warranted in certain scenarios depending on patient risks.
