Do the fellowship. Chief resident is, for the most part, is a waste of time as far as experience goes, it might help add some flourish to you CV if you want to do some leadership stuff later or if you want to do a more competitive fellowship and your application is weak (even then, it is debatable how helpful it will be or not.
In a fellowship, your clinical skills are not going to be dulled to the same degree as they might on the 4th year of chief responsibilities (this is, after all, a glorified secretary). It can open some doors as well for you to have a palliative care fellowship, a geriatrics (population will only get older as time goes by), sleep medicine, etc.
Even people that wish to go into more competitive specialties might end up being better off with another fellowship than with a 4th year of chief. I have no first hand experience on this but I'd take a wild guess that two applicants that are otherwise similar in stats, the one with palliative 1year fellowship has better chances to get into cards than the one with 4th year chief. You can sell your experience dealing patients with terminal diseases (such as advanced cardiomyopathy) as an integration of your pre-existing skills (as a palliative expert, or even as a geriatrics) better than someone else can sell their amazing abilities to schedule 20 residents.
Well that's my opinion anyway, I am sure a bunch of people will get offended by my not so high opinion of chief years, I am sure there are specific cases where it makes sense (in fact one of my collagues did it and it was the best choice at the time in my opinion) but on average, I think it is a waste of time.