Final answer:
The patient's symptoms and medical history point towards a diagnosis of subacute bacterial endocarditis, likely related to bacteria entering the bloodstream during her dental surgery.
Step-by-step explanation:
The patient's history of Tetralogy of Fallot and mitral valve prolapse, combined with symptoms of intermittent fevers, weight loss, fatigue, headaches, skin lesions, hemorrhages in nail beds, and blood cultures growing Gram positive cocci in chains, is highly suggestive of bacteremia and endocarditis. The dental surgery four months prior might have been a significant event despite the absence of immediate complications, as oral flora can enter the bloodstream during dental procedures and potentially infect cardiac structures, particularly in patients with pre-existing heart conditions who did not receive prophylactic antibiotics. Considering all clinical symptoms and the blood culture results, the diagnosis leans towards subacute bacterial endocarditis (SBE), a form of endocarditis that has a more insidious onset and may be less aggressive compared to its acute counterpart.