Does the sentence sound good to you, given the context that I have been spending some time preparing for the test:
I am becoming increasingly happier with the way I write for the IELTS exam.
Thanks!
"The way I write for the IELTS exam" sounds fine. Because you are practicing for the IELTS exam, your writing is done for the purpose of being used on that exam, so that wording is fine. You are happy with your writing with regards to its usage on the exam.
That said, in your sentence I would either change "happier" to "happy" or remove the word "increasingly." "Happier" implies that your happiness is increasing, so you don't need to use both terms. Either say "I am becoming increasingly happy" or "I am becoming happier."