"Can I obtain any FAA document based on my EASA license remotely?"
No, the FAA recently published national policy prohibiting this.
FAA Notice N 8900.689 (20 Mar 2024–20 Mar 2025):
"
6. Action.
a. Determining the Appropriate Use of RT [remote technology].
ASIs, ASTs, and designated examiners must utilize the guidance found in Order 8900.1, Volume 1, Chapter 3, Section 9 to determine whether the use of RT for a specific administrative airman certification function is appropriate. Table 1 below shows the administrative functions that may be conducted using RT, with the exception of foreign pilot certification activities and Aviation English Language Standard (AELS) evaluations. Those functions must be conducted in person."
As you mentioned, a conversion under the Swiss or EU bilateral agreement requires writing an FAA knowledge test which normally can only be done in the United States.
You could convert your EASA PPL to a South African PPL as an intermediate step.