You are close. Using a modified version of your example works just fine (However, see my updates below to understand why this is happening):
gdalsrsinfo -o xml '+proj=longlat +a=3396190 +b=3396190 +no_defs +rf=1' > /tmp/test1;gdalsrsinfo -o wkt /tmp/test1
I'm taking the output (in GML) of the first command and sending it to a temporary file, then reading it back in wkt format. This demonstrates that what you are asking is indeed possible with gdalsrsinfo.
There is an issue with directly attempting to pipe the output from one gdalsrsinfo call to another because of the way it detects SRS information from a string or a file. May be a bug, I don't know.
Also notice I included
+datum=WGS84 to give it something to find in the srs tags. This may be causing you an issue if you were using gml generated from your original test string.
UPDATE: Your issue is being caused because of an inverse flattening value of 0. Add
+rf=1 and tweak to your preferred ellipsoid and you will find that this fixes the error. (I'm using 1 as a placeholder of course). I've updated the command string above.
To get an appropriate value, use the formula
which in this case is 0 for you, but this doesn't compute for gdalsrsinfo.
UPDATE 2: If you change your ellipsoid slightly, this will fix the issue as well without having to fake the inverse flattening value. Here's an updated command string that does what you want:
gdalsrsinfo -o xml '+proj=longlat +a=3396195 +b=3396190 +no_defs' > /tmp/test1;gdalsrsinfo -o wkt /tmp/test1
So we see that it doesn't like perfect spheres!