I would simply suggest writing it with normal markdown, avoiding even HTML. It won't capture the exact appearance, but we're already sacrificing that for a lot of things, like layout and poetry. We already accept that we get paragraphs that don't indent, we can't make a line or word 20% larger, we can only do three levels of bullets (in markdown), etc. and we simply adapt to the layout and formatting that's available.
In this case I would suggest either adopting the normal all-caps appearance:
DAMN. I WISH THEY DIDN'T LEAVE ME TO DEAL WITH THIS SORT OF THING, Death sighed. YOU BELIEVE, BUT YOU DON'T BELIEVE IN ANYTHING.
which is a bit more visually intrusive, but in mixed text not so much, or use bolding:
Damn. I wish they didn't leave me to deal with this sort of thing, Death sighed. You believe, but you don't believe in anything.
which I believe preserves the sense that this speech is more weighty while preserving the ease of readability of mixed-case text.