How can I get recommendation letters for PhD application, while just starting a PhD program?
Seems to me that your best course of action would be to send short emails to your current letter writers. You can thank them again for their letters, but state that, unfortunately, for [personal/medical/visa/whatever] reasons, you'll likely have to leave. (Note that there is no need for all the details! If they're curious they can ask you to elaborate, but the first message from you can indeed be short.) Then ask if you can re-use the letters from the dossier service for the positions that allow for that, or if they'd be willing to send letters directly. If they wrote their letters hoping you'd do well they'd likely agree to both, or, ideally, even to update and adapt the letters for the new positions.

I think that tactic has a better chance of working than convincing new professors to use a dossier service they're not familiar with, and it avoids your letter writers from feeling you went behind their back. Indeed, things could go sour if one of the advisors you apply with were to contact your letter writers to check if everything's above board and cleared with them.