Edit: derp, I didn't read all the words. Yes, you're overdrawing your smart plug with inrush current fron a large inductive load. Look for a smart plug rated for at least 20A. It'll be safe enough to use on a 15A circuit because the breaker will be the limiting factor instead if the plug.

~~Check the amperage and wattage rating on that plug. Heavily inductive and some resistive loads can draw a large "inrush" current during startup that can briefly exceed the capacity of some smart receptacles. Chances are it's rated for 10A resistive load, and a washer (which is an inductive loadl can briefly exceed that.~~

For example: halogen lights. A halogen light tube like you'd use in floodlights or those old floor-standing torchiere are tyoically rated fo 150 to 300W. That rating is derived from the bulb's filament when it is up to ita operating temperature, which is something like 1000°C or more. A hot filament has a higher resistance than that same filament when it's cold, so that 300W element might briefly draw over 600W instantaneous load while it's coming up to temperature, briefly drawing potentially more than 15A at 120V. It doesn't pop your breaker because they have a sort of time delay that can accommodate these brief, large inrush currents. Electric motors in pumps and drives have a similar effect, but that's more from their inductance/capacitance as they spin up, rather than a change in their resistance. Their "apparent resistance" is actually AC inductance, which can draw significantly more amperage when the motor is not spinning at its designed operating speed. Same thing for compressors like in air conditioners, fridges and ice makers.

Most ZigBee smart switches and receptacles are rated for 10A at 120V, or total 600W continuous load. They're designed to handle smaller inrush currents from loads like incandescent bulbs, LEDs, and small electrics. They're generally not rated for things like halogen bulbs or large motors.

Washing machines are an interesting case because they employ several motors to drive the agitator (or drum), the spin cycle and the drain pump. They have a highly dynamic cureeebt draw that depends on both startup inrush, but also on the torque demanded for varying cycles. A heavy load if jeans or towels can draw significantly more power to agitate and spin dry than a lught load of bed linens or delicates.

~~I bet~~ your smart plug is rated for 600W, and probably says not to use on halogen bulbs. (Most do.) If so, its also not going to be suitable for your washer for substantially the same reason: peak draw exceeds its capacity.

For higher draw applications, you'll need to look for a 20A or 30A rated smart switch. They're available on AliExpress at least, and possibly Amazon as well. I can't recommend any for you because I don't use them, myself. Look for ones that are designed for window or wall Air Conditioning units.

Anyway, you're not likely to be able to solve your problem with your current kit. It'll have to be an upgrade or a rethink.

Good luck!