Cooling Methods for Voltage Regulators
Don't stuck into one approach, there are many options available. Don't make yourself difficult.

Use Battery Management System (BMS) + 5V buck dc-dc converter directly on to the 5v rail.
There are tons of it in the online store, some of them are already built-in.
And there are people who wants to create it themselves such as GreatScott!

The BMS will protect your battery and your device from faults, such as protection from overcurrent, short-circuit, overdischarge and so on.

Your main goal is to use a 4S LiPo battery which will have a maximum voltage of 16.8 Volts
Just focus on it. You don't need addional heatsink and fan. Your device for the 200mA load will be cool.