I use the esm version of date-fns and you can use the same type of formats that moment uses :

import { format } from 'date-fns/esm'
import { enUS, fr } from 'date-fns/esm/locale'

I'll store the locales in an object :

this.dateLocales = { fr: fr, en: enUS }

and use these formats :

LT: 'h:mm aa',
LTS: 'h:mm:ss aa',
L: 'MM/DD/YYYY',
LL: 'MMMM D YYYY',
LLL: 'MMMM D YYYY h:mm aa',
LLLL: 'dddd, MMMM D YYYY h:mm aa'

So you can do :

format(
  new Date(2014, 6, 2),
  'LL',
  {locale: this.dateLocales.fr}
)

Those formats are localised

As you are using the FP version to format a date you need to use the formatWithOptions function to pass the locale options.

import format from "date-fns/fp/formatWithOptions";

It has a slightly different function signature then the format function, where you need to pass the options as the first argument.

const formatMonth = format({ locale: de }, "MMM");

Here's the code I am using for doing dynamic lookups using Expo's Localization object.

import * as Localization from 'expo-localization';
import * as Locales from 'date-fns/locale';
import { Locale } from 'date-fns';

/**
 * Looks up a date-fns locale from the Expo localization object.  This falls back to `en-US`
 * @param localization Expo Localization object containing the locale and region.
 * @returns date-fns locale.
 */
export function getDateFnsLocale({ locale, region }: Pick<typeof Localization, 'locale'|'region'>) : Locale {
  return (
    Locales[locale.substring(0, 2) + region] ?? Locales[locale.substring(0, 2)] ?? Locales.enUS
  );
}

Here's the test

import { enUS, fr, frCA } from 'date-fns/locale';

describe('date-fns locale lookup', () => {
  it('shound find fr', () => {
    expect(getDateFnsLocale({ locale: 'fr', region: null })).toBe(fr);
  });
  it('shound find fr-CA', () => {
    expect(getDateFnsLocale({ locale: 'fr-CA', region: 'CA' })).toBe(frCA);
  });
  it('shound not find zz-ZZ', () => {
    expect(getDateFnsLocale({ locale: 'zz-ZZ', region: 'ZZ' })).toBe(enUS);
  });
});

The P patterns to format deal with localization: https://date-fns.org/v2.25.0/docs/format

Looks like you would want the PP variant and can be used like this:

import { caLocale } from 'date-fns/locale/ca'

const result = format(new Date(2021, 0, 12), "PP", {
  locale: caLocale
})

To be honest, I'm not quite sure how webpack.ContextReplacementPlugin works in this context, but to replace the default locale you can use NormalModuleReplacementPlugin:

new webpack.NormalModuleReplacementPlugin(
  /locale\/en-US/,
  'date-fns/locale/ru'
)

It will replace en-US locale, so you can skip passing ru to functions. I just tested it on my app and works as expected: