While you seem to be asking several questions in this thread, I will try to answer them as well as I can.
Is chronic fatigue syndrome a symptom of myocarditis?
No, chronic fatigue syndrome can not be a symptom of myocarditis because CFS is a specific, separate medical condition that is pathophysiologically understood to be neurological, immunological, or endocrine in nature. Moreover, CFS is a chronic condition, while myocarditis is (mostly) an acute illness. A better question could be: "Are CFS and myocarditis etiologically similar?"
because both these conditions can be induced by Epstein-Barr and enterovirus, and because both are debilitating conditions for which attempts to exercise can actually be harmful, I find myself wondering how much they have in common.
While this is true, there are plenty of other causes for myocarditis (e.g., other viruses, bacteria, autoimmune disease, toxins like alcohol or heavy metals, etc.). The reason why exercise can be harmful in these patients is inherently different. While CFS patients usually just experience a subjective worsening of their symptoms, myocarditis patients suffer from heart failure due to their typically reduced ejection fraction, leading to a potentially fatal arrhythmia.
Possibly TNF-alpha could be involved in the initial stimulus?
It most certainly is. Research has shown that TNF-alpha promotes myocarditis, which comes at no surprise since TNF-alpha is a cytokine that is virtually ubiquitous in all inflammation involving macrophages, T-cells and NK cells.
Could that signal that the heart is damaged and direct a "lock-down" of all activities requiring increased circulation?
There is no real "lock-down" of activities requiring increased circulation in myocarditis; your heart just does not pump enough blood due to the muscle tissue being inflamed. Loss of function ("functio laesa") was identified by Galen as a sign of all inflammation. It is understood to be either a neurological response to pain, or caused by swollen tissue inhibiting movement, or both. All this is not exclusive to myocarditis, but rather the final common path of all inflammation.
I hope this helps to answer your questions!