For AMHR, either party can be a member--the person whose name appears on the work order must be the member, but the other party can be the one to write the check.
For AMHA, the person who sends in/pays for the paperwork must be the member.
When I bought my first Minis, it was late in the fall. I wasn't going to buy my membership until the new year. So, the seller gave me a completed work order (with her name & membership number) along with the completed transfers & the papers, & I wrote out the check & sent it all in. No problem.
We did the same thing for AMHA, but it was no dice with them. Even though seller (who was a member) sent in the paper work, because it was my check & I wasn't a member, I had to either pay the non member fee or buy a membership.
So, if you're a member of both registries & you send in the paperwork & payment, your buyer doesn't need to be a member.
Just to clarify though--you will need to send in the registration applications as well as transfers. I'm not entirely sure if that's how you're meaning to do it? You cannot just show you as the breeder and the buyer as original owner, because the foal must first be registered in the name of the person who owned the dam at time of foaling. From your posts I'm not just sure what you were thinking on this.