I think the only way would be to hardcode the versioning. Laravel mix, when using version() in the mix file, creates a file called mix-manifest.json, which looks something like this:
{
"/js/app.js": "/js/app.js?id=0a761df35372545b9884",
"/css/app.css": "/css/app.css?id=50fdf4b6fbed784ca6de"
}
It's a straight map from the original asset name to the versioned name. You'd just have to manually copy the versioned name into your html (which is what the mix() php function does automatically).
So instead of the traditional
<link href="/css/app.css" rel="stylesheet">
you'd have to look at the mix-manifest.json file and manually use the hashed version for that file, like
<link href="/css/app.css?id=0a761df35372545b9884" rel="stylesheet">
And then you'd have to manually update your html every time you change app.css to use the new hash after you run npm run dev. It kind defeats the automatic versioning system of Laravel Mix if you're not using php on your production server.