Modified version of "BetterStyleBundle" from https://tech.trailmax.info/2014/01/mvc-bundling-and-minification-a-betterstylebundle/

AsIsBundleOrderer.cs

public sealed class AsIsBundleOrderer : IBundleOrderer
{
    private AsIsBundleOrderer()
    {
    }

    public static IBundleOrderer Instance { get; } = new AsIsBundleOrderer();

    public IEnumerable<BundleFile> OrderFiles(BundleContext context, IEnumerable<BundleFile> files) => files;
}

BetterStyleBundle.cs

public sealed class BetterStyleBundle : Bundle
{
    public BetterStyleBundle(string virtualPath) : base(virtualPath)
    {
    }

    public override IBundleOrderer Orderer
    {
        get { return AsIsBundleOrderer.Instance; }
        set { }
    }

    public override Bundle Include(params string[] virtualPaths)
    {
        foreach (string path in virtualPaths)
        {
            Include(path, CssRewriteUrlTransformAppRelatve.Instance);
        }

        return this;
    }
}

CssRewriteUrlTransformAppRelatve.cs

public sealed class CssRewriteUrlTransformAppRelatve : IItemTransform
{
    private static readonly Regex UrlRegex = new Regex("url\\(['\"]?(?<url>[^)]+?)['\"]?\\)", RegexOptions.Compiled | RegexOptions.ExplicitCapture);

    private CssRewriteUrlTransformAppRelatve()
    {
    }

    public static IItemTransform Instance { get; } = new CssRewriteUrlTransformAppRelatve();

    public string Process(string includedVirtualPath, string input)
    {
        if (includedVirtualPath == null) throw new ArgumentNullException(nameof(includedVirtualPath));
        if (string.IsNullOrWhiteSpace(input)) return input;

        string directory = VirtualPathUtility.GetDirectory(includedVirtualPath);
        if (string.IsNullOrWhiteSpace(directory)) return input;
        if (directory[directory.Length - 1] != '/') directory += "/";

        return UrlRegex.Replace(input, RebaseUrl);

        string RebaseUrl(Match match)
        {
            string url = match.Groups["url"].Value;
            if (string.IsNullOrWhiteSpace(url)) return match.Value;
            if (url.StartsWith("data:", StringComparison.Ordinal)) return match.Value;
            if (url.StartsWith("https://", StringComparison.Ordinal)) return match.Value;
            if (url.StartsWith("http://", StringComparison.Ordinal)) return match.Value;
            if (url[0] == '/') return match.Value;

            url = VirtualPathUtility.ToAbsolute(directory + url);
            return "url(\"" + url + "\")";
        }
    }
}