I ended up creating a console application to register the cultures I need:

// Namespaces.
using System;
using System.Globalization;
using System.Linq;

/// <summary>
/// Console application that ensures the necessary cultures exist on the system.
/// </summary>
/// <remarks>
/// This console application was necessary because some systems don't contain the necessary
/// cultures, and this logic couldn't reside within the website as the creation of new
/// cultures requires administrative privileges.
/// </remarks>
class Program
{

    #region Main

    /// <summary>
    /// Main entry point for the console application.
    /// </summary>
    static void Main(string[] args)
    {
        EnsureCulturesExist();
    }

    #endregion

    #region Private Methods

    /// <summary>
    /// Ensures all the necessary cultures exist (e.g., "en-AT" doesn't exist on some computers).
    /// </summary>
    /// <remarks>
    /// Some of this code comes from https://msdn.microsoft.com/en-us/library/system.globalization.cultureandregioninfobuilder.aspx
    /// and http://stackoverflow.com/a/16476935/2052963.
    /// </remarks>
    private static void EnsureCulturesExist()
    {

        // Variables.
        var allCultures = CultureInfo.GetCultures(CultureTypes.AllCultures);
        var ignoreCase = StringComparison.InvariantCultureIgnoreCase;

        // Reusable cultures/regions.
        var english = CultureInfo.CreateSpecificCulture("en-US");
        var austria = new RegionInfo("AT");
        var germany = new RegionInfo("DE");
        var sweden = new RegionInfo("SE");
        var switzerland = new RegionInfo("CH");

        // Register English (Austria) / "en-AT".
        var englishAustriaExists = allCultures.Any(x => "en-AT".Equals(x.Name, ignoreCase));
        if (!englishAustriaExists)
        {
            var englishAustria = new CultureAndRegionInfoBuilder("en-AT", CultureAndRegionModifiers.None);
            englishAustria.LoadDataFromCultureInfo(english);
            englishAustria.LoadDataFromRegionInfo(austria);
            englishAustria.CultureEnglishName = "English (Austria)";
            englishAustria.CultureNativeName = "English (Austria)";
            TryRegisterCulture(englishAustria);
        }

        // Register English (Germany) / "en-DE".
        var englishGermanyExists = allCultures.Any(x => "en-DE".Equals(x.Name, ignoreCase));
        if (!englishGermanyExists)
        {
            var englishGermany = new CultureAndRegionInfoBuilder("en-DE", CultureAndRegionModifiers.None);
            englishGermany.LoadDataFromCultureInfo(english);
            englishGermany.LoadDataFromRegionInfo(germany);
            englishGermany.CultureEnglishName = "English (Germany)";
            englishGermany.CultureNativeName = "English (Germany)";
            TryRegisterCulture(englishGermany);
        }

        // Register English (Sweden) / "en-SE".
        var englishSwedenExists = allCultures.Any(x => "en-SE".Equals(x.Name, ignoreCase));
        if (!englishSwedenExists)
        {
            var englishSweden = new CultureAndRegionInfoBuilder("en-SE", CultureAndRegionModifiers.None);
            englishSweden.LoadDataFromCultureInfo(english);
            englishSweden.LoadDataFromRegionInfo(sweden);
            englishSweden.CultureEnglishName = "English (Sweden)";
            englishSweden.CultureNativeName = "English (Sweden)";
            TryRegisterCulture(englishSweden);
        }

        // Register English (Switzerland) / "en-CH".
        var englishSwitzerlandExists = allCultures.Any(x => "en-CH".Equals(x.Name, ignoreCase));
        if (!englishSwitzerlandExists)
        {
            var englishSwitzerland = new CultureAndRegionInfoBuilder("en-CH", CultureAndRegionModifiers.None);
            englishSwitzerland.LoadDataFromCultureInfo(english);
            englishSwitzerland.LoadDataFromRegionInfo(switzerland);
            englishSwitzerland.CultureEnglishName = "English (Switzerland)";
            englishSwitzerland.CultureNativeName = "English (Switzerland)";
            TryRegisterCulture(englishSwitzerland);
        }

    }

    /// <summary>
    /// Attempts to register the specified culture, swallowing any errors that may occur.
    /// </summary>
    /// <param name="culture">
    /// The culture to register.
    /// </param>
    private static void TryRegisterCulture(CultureAndRegionInfoBuilder culture)
    {
        try
        {
            culture.Register();
        }
        catch
        {
        }
    }

    #endregion

}

Couldn't build it as part of the website, as it needs to be run as an administrator to change some system files.

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