I'm putting these two options in as answers, even though they don't really answer the question (the answer is, “in bash, no”) in case someone else sees this, and to show the method of swapping stdout and stderr, and to have formatting correct (unlike in comments).
Answer one: suppose this is a program you want to run (I know it doesn't look like a command but pretend it is; replace all of this with the command and arguments you actually want to run):
{ echo stdout; echo stderr >&2; }
Now suppose you want to grep only stderr for something, but keep stdout as-is.
{ echo stdout; echo stderr >&2; } 2> >(grep something)
If that's all you did you'd get all of stdout, and parts of stderr that have something in it, all interspersed. You may want to write stdout and stderr to separate files for later viewing:
{ echo stdout; echo stderr >&2; } 2> >(grep something > errorfile) > outputfile
For swapping stdout and stderr (and why you'd want to do that) see https://stackoverflow.com/questions/13299317/io-redirection-swapping-stdout-and-stderr for a better explanation than I can give.
The link @JeffSchaller provides is good, but I don't see the swapping stdout and stderr method. Are you interested in that?
– Larry Apr 03 '20 at 21:19$ { echo stdout; echo stderr >&2; } 2> >(cat -n) stdout 1 stderr
– Larry Apr 03 '20 at 21:24