>> +        (setq-local font-lock-keywords-case-fold-search hi-lock-case-fold-search)
>
> This affects all the font-lock-keywords, so it's likely to mess things
> up for the non-hi-lock keywords.
>
> I think we should change the patterns added to `font-lock-keywords`
> instead, such that they do
>
>     (let ((case-fold-search hi-lock-case-fold-search)) <...>)
>
> around the corresponding regexp search.

I tried this, and it works well.  Then instead of adding defcustom I copied
all related details from occur to highlight-regexp/highlight-symbol-at-point
and from isearch-occur to isearch-highlight-regexp to make occur/hi-lock
identical in regard how they handle case-folding (docstrings were copied too).

There is one remaining case that is unclear - whether to use
case-fold-search in hi-lock-process-phrase.  Its comment says:

    ;; FIXME fragile; better to just bind case-fold-search?  (Bug#7161)

But according to docstring of highlight-phrase:

  When called interactively, replace whitespace in user-provided
  regexp with arbitrary whitespace, and make initial lower-case
  letters case-insensitive, before highlighting with `hi-lock-set-pattern'.

I'm not sure if "make initial lower-case letters case-insensitive"
the same as this code

   (if (and case-fold-search search-upper-case)
       (isearch-no-upper-case-p regexp t)
     case-fold-search)

shared between occur and hi-lock in this patch: