I think that it would be simpler to add a decorator which wraps the result of an asynchronous function into an object which can be awaited more than once:

def reawaitable(func):
    @wraps(func)
    def wrapper(*args, **kwargs):
        return CachedAwaitable(func(*args, **kwargs))
    return wrapper

It can be combined with lru_cache and cached_property any third-party caching decorator. No access to internals of the cache is needed.

@lru_cache()
@reawaitable
async def coro(...):
    ...

@cached_property
@reawaitable
async def prop(self):
    ...