Hello and welcome to JustAnswer!
What you're describing does sound like trapped air in the cooling system. The first thing I would recommend is bleeding the cooling system, which I'm assuming may not have been done when the engine was installed.
With the engine cool remove the radiator cap and add coolant until it is nearly full.
With the cap off start the engine and let it run until it reaches operating temperature. Along the way the level will most likely drop as air comes up, you can periodically top it off if necessary.
When the thermostat starts to open then the level will become unstable, the level will likely come up and it may spit some back, so be careful of the hot coolant and steam and have a drain pan under it just in case.
When the thermostat is staying open and the engine is at operating temperature make sure that you have heat. If not then there is still air trapped in the heater core and hoses, you can raise the RPM's to help force coolant through.
When it's staying at operating temperature, you have heat and no more air is coming up then you can do a final top off of the coolant, install the radiator cap and then drive it to see if that takes care of the overheating and fluctuation.