Create class that acts both as nn.Parameter and as nn.Module

To facilitate managing trainable hyperparameters, I am looking for a way to create a class Hyperparameter that acts both as a nn.Parameter and a nn.Module. In particular, I would like to use Hyperparameter objects both as a nn.Parameter (e.g. for algebraic manipulations) but still have access to the interface provided by nn.Module to for example store the objects in a nn.ModuleDict along with other modules or use methods like zero_grad(), parameters().

I tried to accomplish this through multiple inheritance but I think this might be dangerous:

import torch

class Hyperparameter(torch.nn.Parameter, torch.nn.Module):
    def __new__(cls, tensor, name):
        return torch.nn.Parameter.__new__(cls, data=tensor)

    def __init__(self, tensor, name):
        self.register_parameter(name, self)

hp1 = Hyperparameter(torch.ones(5), "test1")
hp2 = Hyperparameter(torch.ones(8), "test2")

hp_dict = torch.nn.ModuleDict({"hp1": hp1, "hp2": hp2})"cpu"))
# KeyError: "attribute 'data' already exists"

This works for the things I described, but calling to() throws an error. I think something is no longer as nn.Module expects it to be, but I am not sure what it is.