When would it be reasonable to consider enhanced external counter-pulsation therapy in patients with refractory angina despite maximally tolerated anti-anginal therapy?  

I would exhaust all options for coronary revascularization first. 

First I would titrate anti-anginal medications including beta blockers, non-dihydropyridines (as needed/tolerated), nitrates and ranolazine.If still refractory angina then EECP is FDA-approved for refractory angina (Class IIB).Sharma et al., PMID 24510321

With the current capabilities of revascularization and optimization of meds, it is rare and far when we reach for EECP. That being said, I agree that it is a viable FDA-approved option when the standard of medical care and revascularization have been exhausted.

EECP remains an effective therapy for improving refractory angina symptom burden. Optimization of medical therapy remains the first line but can be limited by medication tolerance. Revascularization has improved, specifically the result of CTO PCI, and is also an effective intervention for refractor...