This is going in the wrong direction, so I'll answer by showing two alternatives:
- Share e.g. a strong machine generated password for this with enough randomness to pass as a secret key. The reason to share a password instead of another secret key is because I don't know a well used variant of SSH that uses a secret key.
- To share a private key it seems more logical to encapsulate it, i.e. by using regular encryption. You can always send a public key the other way, validate that it was generated by the other party and then encapsulate the existing key. It is probably easiest to do this with an existing protocol such as PGP, which provides helpful services such as key fingerprinting which can be used for the validation.
Many issues have been indicated when it comes to regenerating the same private key for RSA - indicated by the post you already included.
Basically everything about the key pair generation needs to remain the same for the generation algorithm to remain deterministic. Any significant change in the generation part and the resulting key pair will be different. I have for instance described a more efficient way to retrieve random values in a range. But any update in the prime generation part would also break the algorithm.
I would not expect any key pair generation mechanism to be present that is so well described that it can be used deterministically over different versions of the software. If you still want to agree on a private key then I'd go to ECDSA, where the private key is simply a single value in a range and where the public key can be directly derived from the private key.