It seems unlikely he is the grandson of Peter I; Peter I didn't marry until 1883, after Marko was born. I also can't find a Russian warship Odessa, although there were merchant ships by that name around. Often these stories gain a few additional decorative details as they travel through the family, and if he was orphaned at a young age he might not have been given a full and accurate picture by whoever raised him, so you should question every detail and look to find additional proof.
The "birth certificate" might more accurately be described as a translation of an extract of baptism. It does not say he was born in Split, but in Solin, within presumably the diocese of Split. St Stephen is a church there, but it is not a cathedral. I would prioritise getting an original, untranslated copy of this parish register, to see what information there is in it. There may be some errors in the translated copy, and information may be left out.
I also suggest getting a copy of his marriage certificate, which ought to have details of parents as he gave them at the time - hopefully a first name for his mother! She might turn out to be easier to trace, as I think her name is Jelačić and there was a noble family by that name.
Interestingly, in some early NZ/Australian shipping records he seems to be listed as older than a 1880/1 birth would suggest:
May 1903, "M Aprilovich", aged 28
http://marinersandships.com.au/1903/05/087bal.htmalso December 1903, "Marks Aprilovies" ( think the original is "Aprilovic" or "Aprilovice"), aged 29
https://familysearch.org/ark:/61903/3:1:S3HY-6R49-KMQ?i=2&cc=1609792
