## Step 1: Define the symbolic representation of the problem
Let's denote the number of bear plush toys as $x_1$ and the number of dog plush toys as $x_2$. The objective is to maximize profit, where the profit per bear is $4 and the profit per dog is $5. Therefore, the objective function can be represented as $4x_1 + 5x_2$.

## 2: Identify the constraints
The company has 1000 minutes available, with each bear taking 15 minutes to make and each dog taking 12 minutes to make. This can be represented as $15x_1 + 12x_2 \leq 1000$. Additionally, the company must make at least two times as many bears as dogs, which can be represented as $x_1 \geq 2x_2$. Also, $x_1 \geq 0$ and $x_2 \geq 0$ because the number of plush toys cannot be negative.

## 3: Symbolic representation in JSON format
The symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'bear plush toys'), ('x2', 'dog plush toys')],
    'objective_function': '4*x1 + 5*x2',
    'constraints': [
        '15*x1 + 12*x2 <= 1000',
        'x1 >= 2*x2',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## 4: Convert the problem into Gurobi code
To solve this linear programming problem using Gurobi, we will use the Gurobi Python API.

```python
import gurobi

def solve_plush_toys_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name='x1', lb=0, vtype=gurobi.GRB.CONTINUOUS)  # bear plush toys
    x2 = model.addVar(name='x2', lb=0, vtype=gurobi.GRB.CONTINUOUS)  # dog plush toys

    # Set the objective function
    model.setObjective(4 * x1 + 5 * x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(15 * x1 + 12 * x2 <= 1000)  # time constraint
    model.addConstr(x1 >= 2 * x2)  # bear vs dog constraint

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: x1 = {x1.varValue}, x2 = {x2.varValue}")
    else:
        print("The problem is infeasible")

solve_plush_toys_problem()
```