To tackle this problem, we first need to convert the natural language description into a symbolic representation. This involves identifying the variables, the objective function, and the constraints.

- **Variables**: Let's denote the number of A400 keyboards produced as `x1` and the number of P500 keyboards produced as `x2`.
- **Objective Function**: The business wants to maximize profit. Given that one A400 keyboard yields a $35 profit and one P500 keyboard yields an $80 profit, the objective function can be represented as: `Maximize 35*x1 + 80*x2`.
- **Constraints**:
  - Labour constraint: Producing one A400 keyboard requires 5 hours of labour, and producing one P500 keyboard requires 9 hours. The total labour available is 45 hours per week. This constraint can be represented as: `5*x1 + 9*x2 <= 45`.
  - Demand forecast constraint: The business wants to produce at least three times as many A400 keyboards as P500 ones. This can be represented as: `x1 >= 3*x2`.
  - Non-negativity constraints: The number of keyboards produced cannot be negative, so we have: `x1 >= 0` and `x2 >= 0`.

Now, let's put this into the required symbolic representation format:

```json
{
  'sym_variables': [('x1', 'Number of A400 keyboards'), ('x2', 'Number of P500 keyboards')],
  'objective_function': 'Maximize 35*x1 + 80*x2',
  'constraints': ['5*x1 + 9*x2 <= 45', 'x1 >= 3*x2', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this optimization problem using Gurobi in Python, we'll write the following code:

```python
from gurobipy import *

# Create a new model
m = Model("Keyboard_Production")

# Add variables
x1 = m.addVar(lb=0, name="A400_Keyboards")
x2 = m.addVar(lb=0, name="P500_Keyboards")

# Set the objective function
m.setObjective(35*x1 + 80*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(5*x1 + 9*x2 <= 45, "Labour_Constraint")
m.addConstr(x1 >= 3*x2, "Demand_Forecast_Constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of A400 keyboards to produce: {x1.x}")
    print(f"Number of P500 keyboards to produce: {x2.x}")
    print(f"Maximum profit: ${35*x1.x + 80*x2.x:.2f}")
else:
    print("No optimal solution found.")
```