To solve this optimization problem, we first need to define the symbolic representation of the variables and the objective function, as well as list out all the constraints.

The variables in this problem can be represented symbolically as follows:
- Let \(x_1\) represent the number of plush toys made.
- Let \(x_2\) represent the number of action figures made.

The objective function aims to maximize profit. Given that each plush toy yields a profit of $4 and each action figure yields a profit of $4.50, the objective function can be written as:
\[ \text{Maximize: } 4x_1 + 4.5x_2 \]

The constraints are based on the available time for assembly and packaging:
- Assembly time constraint: \(20x_1 + 15x_2 \leq 1200\)
- Packaging time constraint: \(4x_1 + 5x_2 \leq 900\)
- Non-negativity constraints (since we cannot produce a negative number of items): \(x_1 \geq 0, x_2 \geq 0\)

Thus, the symbolic representation of the problem in JSON format is:
```json
{
    'sym_variables': [('x1', 'plush toys'), ('x2', 'action figures')],
    'objective_function': '4*x1 + 4.5*x2',
    'constraints': ['20*x1 + 15*x2 <= 1200', '4*x1 + 5*x2 <= 900', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we can use the following code:
```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="plush_toys", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="action_figures", lb=0)

# Set the objective function
m.setObjective(4*x1 + 4.5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(20*x1 + 15*x2 <= 1200, "assembly_time")
m.addConstr(4*x1 + 5*x2 <= 900, "packaging_time")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Plush toys: {x1.x}")
    print(f"Action figures: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```