So in your opinion would running a 160 be ok?
A 160 doesn't let the engine warm up as fast as a 180 will, and will have very little effect on normal operating temperature.
A "warmed over" engine is called that for a reason- altered cam lifts and durations, larger carburetors with additional fuel metering, larger valves/ports all add up to one thing: additional fuel and air being ingested by the engine. This is what generates more power. Additional fuel and air being combusted equals more heat being generated. Additional heat means higher operating temperature. To address this, the system capacity needs to be increased to handle the added demands being put on it. Both air and coolant flow, as well as capacity, need to increase in order to dissipate the additional heat being generated. A higher output engine needs to retain some additional heat in order to properly burn the increased fuel & air volumes being forced into it. A colder thermostat does nothing other than open earlier, delaying the engine from reaching optimal operating temperature. Once a thermostat of the same type opens, be it a 160 or a 180, it is open- no difference in flow. At that point a t-stat becomes superfluous- it may as well not exist; it's purpose is done. From that point, what controls your operating temperature is the rest of your cooling system- the radiator, the water pump, the fan(s).
If you think your engine is running too warm, it's time to look into the rest of your cooling system, not try to "band-aid" it with a simple thermostat swap.
And as I said earlier, I don't think a high of 200* in a performance oriented engine is overly concerning.
To put this into perspective, a 2020 Challenger 6.2 putting out 797 hp (that's double the horsepower of your 360 from a similarly-sized engine) runs a 203* thermostat; so that's a
minimum operating temperature once the t-stat opens. Wouldn't surprise me if it runs 15-20* over that normally.