Re-project your polygon to a pacific-centered CRS (I used EPSG:8859), than it is possible to split it. With other CRS ending at the International Date Line (like 4326), I had the same problem as you.

Screenshot: originally one polygon, divided with Split Features in several separate features:

Re-project your polygon to a pacific-centered CRS (the same you use as project CRS: EPSG:3832; or the same I used EPSG:8859) and set the project CRS to the same projection. Than you can split the polygon. 

With other CRS ending at the International Date Line (like 4326), I encountered the same problem as you, as well as when I had different CRS for layer and project (3832 and 8859). See here for the difference between layer- and project CRS.

Screenshot: originally one polygon, divided with Split Features in several separate features. CRS is EPSG:8859 for both layer and project: