Ok, finally figured it out. Definitely was a Django setting. Added the following to the Django settings.py:

FORCE_SCRIPT_NAME = '/api/'

Then I had to update the STATIC_URL because it was no longer serving the assets for the admin portal:

STATIC_URL = '/api/static/`

Full settings.py that I'm using, but really the only things that needed changing were adding FORCE_SCRIPT_NAME and updating STATIC_URL:

Django settings for config project.

Generated by 'django-admin startproject' using Django 2.2.6.

For more information on this file, see
https://docs.djangoproject.com/en/2.2/topics/settings/

For the full list of settings and their values, see
https://docs.djangoproject.com/en/2.2/ref/settings/
"""

import os
from datetime import timedelta

# Build paths inside the project like this: os.path.join(BASE_DIR, ...)
BASE_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))

# Forces Django to not strip '/api' from the URI
FORCE_SCRIPT_NAME = '/api/'

# Quick-start development settings - unsuitable for production
# See https://docs.djangoproject.com/en/2.2/howto/deployment/checklist/

# SECURITY WARNING: keep the secret key used in production secret!
SECRET_KEY = os.environ['SECRET_KEY']

# SECURITY WARNING: don't run with debug turned on in production!
DEBUG = os.environ['DEBUG'] == 'True'

ALLOWED_HOSTS = [
        'localhost',
        '127.0.0.1',
        '192.168.64.7'
]


# Application definition

INSTALLED_APPS = [
    'django.contrib.admin',
    'django.contrib.auth',
    'django.contrib.contenttypes',
    'django.contrib.sessions',
    'django.contrib.messages',
    'django.contrib.staticfiles',

    # Third-party packages
    'rest_framework',

    # Local packages
    'authentication'
]

MIDDLEWARE = [
    'django.middleware.security.SecurityMiddleware',
    'whitenoise.middleware.WhiteNoiseMiddleware',
    'django.contrib.sessions.middleware.SessionMiddleware',
    'django.middleware.common.CommonMiddleware',
    'django.middleware.csrf.CsrfViewMiddleware',
    'django.contrib.auth.middleware.AuthenticationMiddleware',
    'django.contrib.messages.middleware.MessageMiddleware',
    'django.middleware.clickjacking.XFrameOptionsMiddleware',
]

ROOT_URLCONF = 'config.urls'

TEMPLATES = [
    {
        'BACKEND': 'django.template.backends.django.DjangoTemplates',
        'DIRS': [],
        'APP_DIRS': True,
        'OPTIONS': {
            'context_processors': [
                'django.template.context_processors.debug',
                'django.template.context_processors.request',
                'django.contrib.auth.context_processors.auth',
                'django.contrib.messages.context_processors.messages',
            ],
        },
    },
]

WSGI_APPLICATION = 'config.wsgi.application'


# Database
# https://docs.djangoproject.com/en/2.2/ref/settings/#databases

DATABASES = {
    'default': {
        'ENGINE': 'django.db.backends.postgresql',
        'NAME': os.environ['PGDATABASE'],
        'USER': os.environ['PGUSER'],
        'PASSWORD': os.environ['PGPASSWORD'],
        'HOST': os.environ['PGHOST'],
        'PORT': 1423
    }
}

# REST Framework settings
# https://www.django-rest-framework.org/
REST_FRAMEWORK = {
    'DEFAULT_RENDERER_CLASSES': [
        'rest_framework.renderers.JSONRenderer',
    ],
    'DEFAULT_AUTHENTICATION_CLASSES': (
        'rest_framework_simplejwt.authentication.JWTAuthentication',
    ),
    'DEFAULT_PERMISSION_CLASSES': [
        'rest_framework.permissions.AllowAny',
    ],
}

# SIMPLE_JWT Settings
# https://github.com/davesque/django-rest-framework-simplejwt
SIMPLE_JWT = {
    'ACCESS_TOKEN_LIFETIME': timedelta(minutes=5),
    'REFRESH_TOKEN_LIFETIME': timedelta(days=1),
    'ROTATE_REFRESH_TOKENS': False,
    'BLACKLIST_AFTER_ROTATION': True,

    'ALGORITHM': 'HS256',
    'SIGNING_KEY': os.environ['SECRET_KEY'],
    'VERIFYING_KEY': None,
    'AUDIENCE': None,
    'ISSUER': None,

    'AUTH_HEADER_TYPES': ('Bearer',),
    'USER_ID_FIELD': 'id',
    'USER_ID_CLAIM': 'user_id',

    'AUTH_TOKEN_CLASSES': ('rest_framework_simplejwt.tokens.AccessToken',),
    'TOKEN_TYPE_CLAIM': 'token_type',

    'JTI_CLAIM': 'jti',

    'SLIDING_TOKEN_REFRESH_EXP_CLAIM': 'refresh_exp',
    'SLIDING_TOKEN_LIFETIME': timedelta(minutes=5),
    'SLIDING_TOKEN_REFRESH_LIFETIME': timedelta(days=1),
}

# Password validation
# https://docs.djangoproject.com/en/2.2/ref/settings/#auth-password-validators

AUTH_PASSWORD_VALIDATORS = [
    {
        'NAME': 'django.contrib.auth.password_validation.UserAttributeSimilarityValidator',
    },
    {
        'NAME': 'django.contrib.auth.password_validation.MinimumLengthValidator',
    },
    {
        'NAME': 'django.contrib.auth.password_validation.CommonPasswordValidator',
    },
    {
        'NAME': 'django.contrib.auth.password_validation.NumericPasswordValidator',
    },
]


# Internationalization
# https://docs.djangoproject.com/en/2.2/topics/i18n/
LANGUAGE_CODE = 'en-us'
TIME_ZONE = 'America/Los_Angeles'
USE_I18N = True
USE_L10N = True
USE_TZ = True


# Static files (CSS, JavaScript, Images)
# https://docs.djangoproject.com/en/2.2/howto/static-files/
STATIC_URL = '/api/static/'
STATIC_ROOT = os.path.join(BASE_DIR, 'static')

Dockerfile

FROM python:3.7-slim
ENV PYTHONUNBUFFERED 1
WORKDIR /app
EXPOSE 5000
COPY requirements*.txt ./
RUN pip install -r requirements.txt
COPY . .
RUN python manage.py collectstatic
CMD ["gunicorn", "-b", ":5000", "--log-level", "info", "config.wsgi:application"]