This is amazing and makes so much sense to me. I hope at least one person takes up this great idea!

This is an interesting idea, I think your post could be stronger & allow people to evaluate whether they want to participate better if it was more clear on the following points: 

Logistics related: 

• Is "we" referring to Overcome? How is your organization involved in this project? You mention you provide fiscal sponsorship but it might be worth first spelling out who you are, your track record and whether someone would want to work / volunteer for your organization. 
  • You say "We've got 3.5 years of experience running EA mental health projects with volunteers." - what kinds of projects have you run in the past (e.g. aside from those you have posted about in the past year?)
  • What has been the outcome of the experiments that Overcome has launched on the EA Forum in the past year?
• Is there a reason why this position is unpaid? I'd be surprised if there are many people who could spend 150+ hours on an unpaid project over a few months. I'd guess if this idea seems relatively cost-effective (e.g. through a BOTEC) you could likely get a some money to pay someone to figure out if it's worth the time.  
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Intervention related: 

• Why this intervention, as opposed to others? There are a lot of things someone could do to improve mental health in developing countries, it's not clear that going for complex cases is a resource-competitive intervention to focus on (as opposed to guided self help which scales a lot more, even if it may be for simpler cases)
  • If you have a BOTEC or more details on the implementation, it would be good to share those.
• You write: "A substantial minority of Western therapists would happily help them pro bono via Zoom." - is this based on your experience? a study?
• You don't talk a lot about the client-side matching - which seems pretty important for success 
  • Do you already have buy-in from mental health organizations which would provide the supply of clients from the low-income countries? 
  • Have you explored potential barriers in therapist matching e.g. cultural and language barriers? 

Quick question - my wife is a provisionally licensed LPC in the US. I know there's a lot of rules on how they are allowed to practice (like they must be with a client within their state of licensure). Do these rules just not apply when working internationally?