Thank you very much for your reply @Clubdebambos  and for going as in-depth as you did!
To clarify, I have been attempting to set showLabels within the feature layer and not within the service. However, I have been using a slightly different technique than what you have done.
from arcgis import features, GIS
# Get the feature service item
item = agol.content.get("Feature Service ID")
# Convert to FeatureLayerCollection object
flc = features.FeatureLayerCollection.fromitem(item)
# Add/append layer definitions
flc.manager.add_to_definition({"layers": [layer_definitions]})
 
Each definition in layer_definitions is roughly as follows
{
    "type": "Feature Layer",
    "extent": {
        "xmax": 180.0,
        "ymin": -90.0,
        "xmin": -180.0,
        "ymax": 90.0,
        "spatialReference": {
            "wkid": 4326,
            "latestWkid": 4326
        }
    },
    "hasAttachments": false,
    "name": <layer_name>,
    "fields": [
        {
            "alias": <field_alias>,
            "name": <field_name>,
            "type": "esriFieldTypeString"
        }
    ],
    "drawingInfo": {
        "renderer": {
            "type": "simple",
            "symbol": {
                "type": "esriSMS",
                "color": [
                    66,
                    135,
                    245,
                    255
                ],
                "style": "esriSMSCircle",
                "size": 10,
            }
        },
        "scaleSymbols": false,
        "labelingInfo": [
            {
                "labelExpressionInfo": {
                    "expression": "$feature[\"<name>\"]"
                },
                "labelPlacement": "esriServerPointLabelPlacementCenterRight",
                "minScale": 0,
                "maxScale": 0,
                "symbol": {
                    "color": [
                        51,
                        51,
                        51,
                        255
                    ],
                    "font": {
                        "decoration": "none",
                        "family": "Arial",
                        "size": 8,
                        "style": "normal",
                        "weight": "normal"
                    },
                    "haloColor": null,
                    "haloSize": null,
                    "horizontalAlignmen": "center",
                    "kerning": true,
                    "rightToLeft": false,
                    "type": "esriTS",
                    "verticalAlignment": "baseline",
                    "xoffset": 0,
                    "yoffset": 0
                },
                "useCodedValues": false
            }
        ],
        "transparency": 0
    },
    "showLabels": true,
    "geometryType": "esriGeometryPoint"
}
 
Every other parameter works as expected, except for showLabels. I end up with a bunch of points that have no labels. If I go into the layer settings on the web map and click "Create Labels", then they show up with the information specified in labelingInfo.
I was going to try to use your method, but I am running into a slight issue. When I use get_data() on the item returned from the feature service, it is returning an empty dictionary. I don't know if that could indicate a larger issue with my creation of the feature service. Additionally, I can see in the returned item object that 'layers' is None despite the fact that I can see my created layers when viewing the map in the portal. Do either of these issues indicate a problem with my setup of the service and it's layers? If yes, could this be causing the labels to not render properly? Given that everything else is working as intended, I believe I set it up correctly, but perhaps I am mistaken.
Thanks again for your time!