It sounds like you're looking for a verb/phrase which can be used in a passive voice (which often sounds more formal and detached). In this sense, the direct equivalent of "under construction" for writing would be "being written":
My book is being written at the moment.
However, since people don't usually use this sort of construction regarding something they're writing themselves, this could run the risk of sounding like "your" book is actually being written by somebody else. Another phrase which sounds a bit less like this might be "in the process of being written":
My book is in the process of being written.
But you might want to be careful because using "in the process" in a passive voice like this can sometimes also give the impression that it's a long process which may not be completed any time soon (which is good if that's what you actually mean, but not if it isn't).
If that's not what you want, then perhaps the best choice may be a combination of "in the process" but used in an active voice, which gives it more immediacy, but is still not as blunt as the original:
I am in the process of writing a book.
Regarding the other possibilities you mentioned, you are correct that "under writing" is not commonly used for this and is likely to be mistaken for "underwriting" (one word), which means something very different (having to do with financial contracts, etc). "Under composition" would actually also be a technically correct option, but may sound a bit strange (or overly formal or dramatic) to some.
