How to integrate `huggingface.PyTorchModelHubMixin.save_pretrained()` with Lightning `Trainer` (checkpointing & loading)

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### Description & Motivation

## 📝 Summary

I am trying to train a custom policy module using **PyTorch Lightning**, where my model components (`policy`, `preprocessor`, `postprocessor`) all inherit from `huggingface_hub.PyTorchModelHubMixin`.

This mixin provides:
- `_save_pretrained()`
- `_from_pretrained()`
which work similarly to HuggingFace’s `save_pretrained()` and `from_pretrained()`, and are very convenient for packaging model weights + config.

However, I am not sure how to properly integrate these HF-style save/load utilities into Lightning's standard training flow — especially Lightning's `ModelCheckpoint` callback.

## 📦 Minimal example

```python
class PolicyModule(LightningModule):
def __init__(
self,
policy: BasePolicy,
preprocessor: DataProcessorPipeline,
postprocessor: DataProcessorPipeline,
**kwargs,
):
super().__init__()

self.save_hyperparameters(
logger=False, ignore=["policy", "preprocessor", "postprocessor"]
)

self.policy = policy
self.preprocessor = preprocessor
self.postprocessor = postprocessor

def training_step(self, batch, batch_idx):
batch = self.preprocessor(batch)
loss, loss_dict = self.policy(batch)

self.log_dict(
{"train/_loss": loss},
on_step=True, on_epoch=True, prog_bar=True, sync_dist=True,
)
self.log_dict(
{f"train/{k}": v for k, v in loss_dict.items()},
on_step=True, on_epoch=True, sync_dist=True,
)
return loss
```

## ❗The problem
### How to make Lightning call `save_pretrained()` during checkpointing?

## 🙏 Additional context
PyTorch Lightning provides many excellent callback tools—especially `ModelCheckpoint`—which greatly simplify training workflows. Ideally, I would like to remain fully within the standard Lightning Trainer + callbacks framework, while still taking advantage of HuggingFace-style `save_pretrained()` / `from_pretrained()` for model components.

Thanks in advance for any guidance or best practices!

### Pitch

_No response_

### Alternatives

_No response_

### Additional context

_No response_

cc @lantiga

Start by reviewing Lightning's Trainer and ModelCheckpoint callback flow alongside the PyTorchModelHubMixin save_pretrained() and from_pretrained() methods described in the issue. Determine whether the expected outcome is a documented integration pattern or a framework change, and define done as a clear checkpointing and loading workflow for the listed model components.

Written by the indexing model from the issue text.