Is the word "throwee" acceptable?

Solution 1:

Neither the OED nor the Corpus of Contemporary American English nor the British National Corpus has any record of throwee, but a Google search produces some 40,700 pages containing it. Use it only if you think your intended readers will understand it. Many general readers, I suspect, would not.

Solution 2:

In the context of programming languages, you are referring to an Exception. "Exception" is a widely-understood term; "throwee" is unacceptable as a synonym.

If you are concerned that it's still not clear enough, "Exception object" or "thrown object" is unmistakably clear.