## Step 1: Define the symbolic variables
Let's define the symbolic variables as follows:
- $x_1$ represents the amount of oil paintings produced (in kg)
- $x_2$ represents the amount of acrylic paintings produced (in kg)
- $x_3$ represents the amount of watercolor paintings produced (in kg)

## Step 2: Formulate the objective function
The profit per kg for selling the paint is $150, $180, $220 for the oil painting, acrylic painting, and watercolor painting respectively. The objective function to maximize profit is:
\[ \text{Maximize:} \quad 150x_1 + 180x_2 + 220x_3 \]

## 3: Define the constraints
The constraints based on the given resources are:
- Dye constraint: $6.5x_1 + 8x_2 + 16x_3 \leq 350$
- Filler constraint: $15x_1 + 12x_2 + 5x_3 \leq 250$
- Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0, x_3 \geq 0$

## 4: Symbolic representation of the problem
The symbolic representation is:
```json
{
    'sym_variables': [('x1', 'oil paintings'), ('x2', 'acrylic paintings'), ('x3', 'watercolor paintings')],
    'objective_function': '150*x1 + 180*x2 + 220*x3',
    'constraints': [
        '6.5*x1 + 8*x2 + 16*x3 <= 350',
        '15*x1 + 12*x2 + 5*x3 <= 250',
        'x1 >= 0',
        'x2 >= 0',
        'x3 >= 0'
    ]
}
```

## 5: Gurobi code
Now, let's implement this in Gurobi using Python:
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="oil_paintings", lb=0, ub=None)
    x2 = model.addVar(name="acrylic_paintings", lb=0, ub=None)
    x3 = model.addVar(name="watercolor_paintings", lb=0, ub=None)

    # Define the objective function
    model.setObjective(150 * x1 + 180 * x2 + 220 * x3, gurobi.GRB.MAXIMIZE)

    # Define the constraints
    model.addConstr(6.5 * x1 + 8 * x2 + 16 * x3 <= 350, name="dye_constraint")
    model.addConstr(15 * x1 + 12 * x2 + 5 * x3 <= 250, name="filler_constraint")

    # Optimize the model
    model.optimize()

    # Print the results
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Oil paintings: {x1.varValue} kg")
        print(f"Acrylic paintings: {x2.varValue} kg")
        print(f"Watercolor paintings: {x3.varValue} kg")
        print(f"Maximum profit: ${model.objVal:.2f}")
    else:
        print("No optimal solution found.")

solve_paint_manufacturer_problem()
```