>> I wonder if it is realistic to use a temporary helper buffer to
>> implement ^.
>
> Good idea.  Maybe even not a buffer, but just a string.
> Are there any differences between buffer matching and string matching?
>
> Then first we could remove ^ from the search regexp, and when it finds
> something, then get the found buffer-substring using text properties
> and match it with the original regexp that contains ^.

This works surprisingly well.  Maybe there are more corner cases,
but something already works with quick tests: