It’s important to prioritize safety and compliance with electrical codes when installing a ceiling fan. A standard ceiling fan bracket typically requires a compatible electrical box that can support its weight and the motion of the fan. Using a 3-inch electrical box with a larger 4-inch bracket is not advisable as it may not provide the necessary support and could lead to safety hazards.
You have a couple of options. One is to replace the existing electrical box with a larger, fan-rated box that fits the fan's bracket. This is the safest and most compliant solution. Fan-rated boxes typically cost between $15 to $50, depending on the type and location, but installation by a licensed electrician may add $100 to $200 to the total cost.
If replacing the box isn't an option, you could consider using a fan brace that can be installed between the ceiling joists to support the fan. However, this still requires a compatible electrical box.
I recommend getting 2-3 local quotes on Handyman.com to discuss your options with licensed professionals who can ensure the installation is safe and up to code.
— Assisted answer