Hold it down with bird netting? Put up a small "silt fence"

It's going to float, and depending on water flow, may end up downstream.

When it really rains, all my wood chip mulch floats. It doesn't matter what kind of wood it is, or how old and broken down it is, it ends up washed up into mini-dunes of wood chips. Where I use old leaves it stays put better but even that will wash away with heavy rain.

Like the idea of bird netting over whatever mulch is selected but netting would have to be easy to lift/remove to weed and then put back in place. Weeds will come and in a stormwater swale you don't want to spray herbicides.

We use stone material over landscaping fabric where the moisture retention and slow-release fertilizing properties of wood mulch are not desirable - for instance under lavender shrubs.

Depending on the stone, it can add some highlighting accent (white pebbles or marble chips, gray crushed rock) or mimic roughly the color of mulch (dark red lava chips).

Maybe stone would work for this application?

No matter where I have bird netting, wildlife of the reptile kind get tangled up in it. Even if it is just kept on a roll in the garage awaiting use, I will find dead snakes, lizards and box turtles tangled up in it. I have to check it when it's draped over the blueberries just to rescue anyone before they get so tangled up that they strangle. Good for keeping birds off of things but somehow bad for reptiles.

Lojon,
Water will move a house if there's enough of it.
I would plant some grass, mabey sod, or baby ornemental grasses, or lirope, and hand water til it takes.
You don't want weeds, so if you plant grass or something else, there won't be any weeds, just a ground cover.
Lirope will even put up with dry conditions probably better than grass.
Just buy the clumping kind of lirope, make sure it dosn't say monkey grass on the container, there are two different kinds.
You want the clumping kind only.
Mine have pretty pink and blue flowers on them.

liriope spicata is the one to avoid unless you want a rampant spreader.

we use river rock in a similar area and it works well.

Most wood mulches are just going to float off the first time the area gets covered in water.