I think standardizing a file descriptor interface would be ideal. It's ridiculous that the file descriptors are exposed and densely packed. I wonder if a protocol could be made that demultiplexes for stderr and the new file descriptor could be made.
But trying to add a fourth now, would likely break too many things; some software will assume any fd > 2 is free for it to clobber.
The IBM mainframe operating system z/OS (formerly MVS), classically instead of numbers for inherited descriptors, the API uses names (DDNAMEs)-that would have made adding a new one a lot easier. But decades too late for that in Unix land, and eventually MVS added the Unix way too for Unix compatibility-the classic API still uses DDNAMEs, but many apps now use the file descriptor-based z/OS Unix API instead