A capacitor wouldn't help in this case. They're most useful when you need to overcome transient situations like a locked rotor, not when you have a heavy drum that needs to be spun up from zero. During this phase the machine can consume well over 2kW of power for a few seconds. The circuit is rated for continuous 15A, but it has no problem supplying 20 or 25 for a brief period of time.

I'm sure that home mains would have no problem, but a solar installation might. Would a small li-ion help smooth over those amperage hills?

It would help at noon, when usually inverters make so much power they push the local grid to saturation which causes them to disconnect from the grid for a while. There are different rules for this for different localities, typically the installer will pick the regime appropriate for the local grid. Where I live (NL) the voltage is allowed to hit 253V for up to a minute, if it takes longer than that my inverters will switch off. That's a requirement to pass inspection, if your inverter can not do this then that's an automatic fail.

There are 'home batteries' that store power for practical reasons up to a day, maybe two. They are useful for sunny/cloudy conditions to maximize the amount of power locally consumed. But battery storage and retrieval isn't free and so far - at least, for my installation - this does not make economic sense. But another factor of two in price drop for my preferred chemistry and it definitely would make sense and then I would probably install about 20KWh worth of capacity. The trick then becomes to balance power sent to the grid and sent to the battery so that you maximize the utility of the grid in order not to wear out the battery prematurely. That means cycling the battery between 70 and 90% state-of-charge for Lithium-Ion. I'd much rather have Lithium Titanate so that is what I'll be holding out for, they are already available, but still too expensive. But they're much safer and have far better charge/discharge curves and life-span.

I actually meant a small li-ion inside the appliance, to smooth out the load. Do any batteries support 20C or 30C? That seems like enough to smooth such a load.

That's a very complex and potentially dangerous device.

I see, thanks.

The problem would not even be making it initially, the problem would be to keep it safe in the long run. I am not a huge fan of all these embedded Lithium-Ion batteries, they're flooding the market (and our homes) and they are disproportionally large factor in house fires.

I did not realize that was so common. A friend of mine actually had a lithium ion fire in his house, from one of those hoverboard devices.

Don't bother; at that point just plug them into 200~300 V DC right to the battery (with just fuse and meter between, no need for any conversion, the washer motor controller that'd be needed to utilize a capacitor for smoothing these spikes would do just fine on a reasonable battery charge/discharge voltage spread.

Do any home battery systems, or home appliances, actually support this?

Most electronic power supplies for computers and similar devices (like e.g. modern TV) have zero complaints about being fed such DC.

Sadly only those that try to sell to Alirack applications (i.e., Chinese server PSU's like for example Lenovo Servers) bother testing in QA and doing enough design review to officially claim that support on their data plate.

You can tear down a sacrifice from a series of identical consumer PSUs though, and reverse engineer the front end schematic to check that the AC side of the active power factor correction design that you'd find in e.g. a decent gaming PSU doesn't do anything that would react badly to a lack of AC.