It's doable simply by not using hyphenation – only justification – and manually inserting word breaks using the space bar, which triggers the automatic justification to rewrap the text line. It can be quite laborious, but that's just how it is.
Just like when you manually correct the automatic hyphenation here and there, it's important to do the manual word break as the last step after all other settings are done. Work from top to bottom, and from right to left, to prevent the auto-justification to wrap finished lines again. The Single-Line Composer (corner menu of the Paragraph panel) is better to work with here because the All-Line Composer would just rewrap all lines. (Anyway, keep a copy of the unaltered text so that you can quickly start over again.)
Before starting the word breakup: use the Justification settings (via corner menu in the Paragraph panel). The goal is to balance out the justification to avoid large gaps between words and letters as good as possible, so that the text area has an even 'grey'. It can take some time to find the optimal setting…
(The skull is a compound shape with holes – it was placed over the text box and set to wrap the text
Object > Text Wrap > Make. In Text Wrap Options, offset value set to 0.) Words have been broken up using spaces to fill gaps so that the shape is.)