The question here is whether the subject predicate in the that-clause is the full noun bloodshed or the noun blood postmodified by the past participle shed (which would be equivalent to “shedding of blood”). The 1920 LDS edition changed the spelling to bloodshed to guarantee the reading as a full noun.
There are 26 occurrences of bloodshed(s) in the Book of Mormon. In at least two more of these cases, one could reinterpret the word bloodshed as the noun blood postmodified by the past participle shed:
In both these cases, unlike Mormon 1:12, there has been no spelling variation; we get only bloodshed.
The Book of Mormon text has many instances of the gerundive expression “(the) shedding (of) blood” (with 28 occurrences). In other words, there are two nominal forms: the full noun bloodshed(s) and the gerundive “(the) shedding (of) blood”. There doesn’t seem to be one clear case of blood postmodified by shed. For this reason, the 1920 emendation of blood shed to bloodshed in Mormon 1:12 was probably correct.
Summary: Maintain the noun bloodshed in Omni 1:24, Helaman 4:1, and Mormon 1:12; the expression blood shed does not seem to occur in the text; instead, the text uses the full noun bloodshed(s) or the gerundive expression “(the) shedding (of) blood”.