Here’s a clean, production-ready refactoring following your target structure. Each module has a single responsibility, imports are properly scoped, and the public API is explicitly controlled in `__init__.py`.

### `queue/__init__.py`
```python
"""Public API for the job queue package."""

from .core import Job, JobQueue
from .priority import PriorityQueue
from .retry import process_with_retry

# Expose exactly what the public API should be
__all__ = ["Job", "JobQueue", "PriorityQueue", "process_with_retry"]
```

### `queue/core.py`
```python
"""Core data structures and FIFO queue operations."""

from dataclasses import dataclass, field
from typing import Any, List, Optional
import uuid
from datetime import datetime


@dataclass
class Job:
    """Represents a unit of work in the queue."""
    id: str = field(default_factory=lambda: str(uuid.uuid4()))
    payload: Any = None
    status: str = "PENDING"
    priority: int = 0  # Higher value = higher priority
    retries: int = 0
    max_retries: int = 3
    created_at: datetime = field(default_factory=datetime.now)


class JobQueue:
    """Basic FIFO job queue implementation."""
    
    def __init__(self) -> None:
        self._jobs: List[Job] = []

    def enqueue(self, job: Job) -> None:
        """Add a job to the end of the queue."""
        self._jobs.append(job)

    def dequeue(self) -> Optional[Job]:
        """Remove and return the next job, or None if empty."""
        return self._jobs.pop(0) if self._jobs else None

    def size(self) -> int:
        """Return the number of jobs in the queue."""
        return len(self._jobs)

    def get_job(self, job_id: str) -> Optional[Job]:
        """Retrieve a job by its ID without removing it."""
        return next((j for j in self._jobs if j.id == job_id), None)
```

### `queue/priority.py`
```python
"""Priority-aware queue extension."""

from typing import Optional
from .core import Job, JobQueue


class PriorityQueue(JobQueue):
    """Queue that processes higher-priority jobs first."""
    
    def enqueue(self, job: Job) -> None:
        """Insert job and maintain priority order (highest first)."""
        super().enqueue(job)
        # In production, consider using `bisect` or `heapq` for O(log n) insertion
        self._jobs.sort(key=lambda j: j.priority, reverse=True)

    def dequeue(self) -> Optional[Job]:
        """Return the highest-priority job."""
        return super().dequeue()
```

### `queue/retry.py`
```python
"""Retry logic with exponential backoff."""

import time
from typing import Callable, Any, Type, Tuple, Optional
from .core import Job


def process_with_retry(
    job: Job,
    func: Callable[..., Any],
    max_retries: Optional[int] = None,
    base_delay: float = 1.0,
    retryable_exceptions: Tuple[Type[Exception], ...] = (Exception,)
) -> Any:
    """
    Execute a function with a job's payload, retrying on failure with exponential backoff.
    
    Updates the job's status and retry count automatically.
    """
    max_retries = max_retries if max_retries is not None else job.max_retries
    last_exception = None

    for attempt in range(max_retries + 1):
        try:
            result = func(job.payload)
            job.status = "COMPLETED"
            return result
        except retryable_exceptions as e:
            last_exception = e
            job.retries = attempt + 1
            job.status = "RETRYING"
            
            if attempt < max_retries:
                delay = base_delay * (2 ** attempt)
                time.sleep(delay)

    job.status = "FAILED"
    raise last_exception
```

### Key Design Decisions:
1. **Single Responsibility**: 
   - `core.py` owns the `Job` dataclass and basic FIFO mechanics
   - `priority.py` extends `JobQueue` to handle ordering logic
   - `retry.py` isolates execution, backoff, and state mutation
2. **No Circular Imports**: All cross-module dependencies point to `core.py`, which has no package-relative imports.
3. **Clean Public API**: `__init__.py` explicitly controls what's available via `import queue`. Secondary classes/functions can still be imported directly if needed.
4. **Type Safety & Maintainability**: Full type hints, dataclasses for state management, and explicit status transitions make the codebase testable and IDE-friendly.
5. **Extensibility**: `PriorityQueue` inherits rather than duplicates logic. `process_with_retry` accepts custom exception tuples and delay multipliers for advanced use cases.

You can now use it like:
```python
from queue import Job, JobQueue, PriorityQueue, process_with_retry

q = PriorityQueue()
q.enqueue(Job(payload={"task": "send_email"}, priority=5))
q.enqueue(Job(payload={"task": "generate_report"}, priority=2))

job = q.dequeue()
process_with_retry(job, func=lambda p: print(f"Processing {p}"))
```