A Raspberry Pi Zero is $5.
SG90 servos are $1 on AliExpress.
A separate power supply is recommended for the Raspberry Pi so that's what you can buy with the rest of the budget.
As far as I can tell, the real ones have four screws in the base, and "TowerPro" printed into the plastic on the top. The fakes have two screws, no brand name moulded into the plastic, and inevitably stop working quite swiftly.
https://brachiograph.readthedocs.io/en/latest/explanation/ha...
Where do you normally order from to guarantee originals?