## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the number of policeman costumes.
- $x_2$ represents the number of fireman costumes.

## Step 2: Formulate the objective function
The profit per policeman costume is $8, and the profit per fireman costume is $10. The objective is to maximize profit, so the objective function is:
\[ \text{Maximize:} \quad 8x_1 + 10x_2 \]

## 3: Define the constraints
1. The store has a budget of $3000, with each policeman costume costing $10 and each fireman costume costing $15. Therefore, the budget constraint is:
\[ 10x_1 + 15x_2 \leq 3000 \]
2. The monthly demand for both costumes will not exceed 280. So, the demand constraint is:
\[ x_1 + x_2 \leq 280 \]
3. Non-negativity constraints, as the number of costumes cannot be negative:
\[ x_1 \geq 0, \quad x_2 \geq 0 \]

## 4: Symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'policeman costumes'), ('x2', 'fireman costumes')],
'objective_function': '8*x1 + 10*x2',
'constraints': [
    '10*x1 + 15*x2 <= 3000',
    'x1 + x2 <= 280',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 5: Gurobi code to solve the problem
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(lb=0, name="Policeman_Costumes")
    x2 = model.addVar(lb=0, name="Fireman_Costumes")

    # Define the objective function
    model.setObjective(8*x1 + 10*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(10*x1 + 15*x2 <= 3000, name="Budget_Constraint")
    model.addConstr(x1 + x2 <= 280, name="Demand_Constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Policeman costumes: {x1.varValue}")
        print(f"Fireman costumes: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("The problem is infeasible")

solve_costume_problem()
```