I figured it out!
Cheers to ilvar for pointing me in the right direction. My implementation is below. I created a property named cache_key and added a post_save receiver onto the sub class of the models whose view-level caches I needed to clear after they had been updated. Suggestions for improvement are always welcome!
from django.conf import settings
from django.core.cache import cache
from django.db.models.signals import post_save
from django.http import HttpRequest
from django.utils.cache import _generate_cache_header_key
from someapp.models import BaseModelofThisClass
class SomeModel(BaseModelofThisClass):
    ...
    @property
    def cache_key(self):
        # Create a fake request object
        request = HttpRequest()
        # Set the request method
        request.method = "GET"
        # Set the request path to be this object's permalink.
        request.path = self.get_absolute_url()
        # Grab the key prefix (if it exists) from settings
        key_prefix = settings.CACHE_MIDDLEWARE_KEY_PREFIX
        # Generate this object's cache key using django's key generator
        key = _generate_cache_header_key(key_prefix, request)
        # This is a bit hacky but necessary as I don't know how to do it
        # properly otherwise. While generating a cache header key, django
        # uses the language of the HttpRequest() object as part of the
        # string. The fake request object, by default, uses
        # settings.LANGUAGE_CODE as it's language (in my case, 'en-us')
        # while the true request objects that are used in building views
        # use settings.LANGUAGES ('en'). Instead of replacing the segment
        # of the string after the fact it would be far better create a more
        # closely mirrored HttpRequest() object prior to passing it to
        # _generate_cache_header_key().
        key = key.replace(settings.LANGUAGE_CODE, settings.LANGUAGES[settings.DEFAULT_LANGUAGE][0])
        return key
    @receiver(post_save)
    def clear_cache_for_this_item(sender, instance, **kwargs):
        # If this is a sub-class of another model
        if sender not in BaseModelofThisClass.__subclasses__():
            return
        else:
            cache.delete(instance.cache_key)