Can you say “hyperbolizing”?
Yes, you can, but it would always be being done "for effect". The usage would be well enough understood by many people but it is extremely unusual.
To do so in a formal context would be unwise as what my be an attempt at humour could well be mistaken for an unsuccessful attempt to appear erudite.
A Google search returns the following results:
hyperbolizing - 24,300 hits
hyperbolising - 12,300 hits
both counts indicate minor use only, but enough to suggest a reasonable level of understanding of the term if it was used.
Google N Gram - ing is useful, but adding a few more comparison terms adds perspective.
Adding "overstating" and "exaggerating" buries "hyperbolizing" and "being hyperbolic" in the noise. ie the use may have increased in absolute terms, in recent decades, but it is still very rare. It would be interesting to know what influence using Google Books as a reference source has on the result.