Unfortunately, labeling rules in ArcGIS Pro cannot translate seamlessly to vector tiles, as long as they are using Mapbox Style Specification .
In fact, according to this specification, there is simply no way to translate a large part of the labeling behaviour provided by the Maplex engine (for instance, stacking, weights, or dynamic positioning) into something that will work in a Vector Tile layer, and in my experience even simple properties that should work properly are not always carried over by the ESRI tools
To fix some of this issues you may have to manually edit the style .json in your hosted server, but this will involve some trial and error if you are not already familar with the Mapbox Specs.
Short answer: don't expect labels to behave in the same way in vector tiles, and stick as much as possible to basic properties.