- You can put tiles into assets folder (if it is acceptable for the app size) or download them all on first start and put them into device storage (SD card). 
 
 - You can implement TileProvider like this: 
 
public class CustomMapTileProvider implements TileProvider {
    private static final int TILE_WIDTH = 256;
    private static final int TILE_HEIGHT = 256;
    private static final int BUFFER_SIZE = 16 * 1024;
    private AssetManager mAssets;
    public CustomMapTileProvider(AssetManager assets) {
        mAssets = assets;
    }
    @Override
    public Tile getTile(int x, int y, int zoom) {
        byte[] image = readTileImage(x, y, zoom);
        return image == null ? null : new Tile(TILE_WIDTH, TILE_HEIGHT, image);
    }
    private byte[] readTileImage(int x, int y, int zoom) {
        InputStream in = null;
        ByteArrayOutputStream buffer = null;
        try {
            in = mAssets.open(getTileFilename(x, y, zoom));
            buffer = new ByteArrayOutputStream();
            int nRead;
            byte[] data = new byte[BUFFER_SIZE];
            while ((nRead = in.read(data, 0, BUFFER_SIZE)) != -1) {
                buffer.write(data, 0, nRead);
            }
            buffer.flush();
            return buffer.toByteArray();
        } catch (IOException e) {
            e.printStackTrace();
            return null;
        } catch (OutOfMemoryError e) {
            e.printStackTrace();
            return null;
        } finally {
            if (in != null) try { in.close(); } catch (Exception ignored) {}
            if (buffer != null) try { buffer.close(); } catch (Exception ignored) {}
        }
    }
    private String getTileFilename(int x, int y, int zoom) {
        return "map/" + zoom + '/' + x + '/' + y + ".png";
    }
}
And now you can use it with your GoogleMap instance:
private void setUpMap() {
    mMap.setMapType(GoogleMap.MAP_TYPE_NONE);
    mMap.addTileOverlay(new TileOverlayOptions().tileProvider(new CustomMapTileProvider(getResources().getAssets())));
    CameraUpdate upd = CameraUpdateFactory.newLatLngZoom(new LatLng(LAT, LON), ZOOM);
    mMap.moveCamera(upd);
}
In my case I also had a problem with y coordinate of tiles generated by MapTiler, but I managed it by adding this method into CustomMapTileProvider:
/**
 * Fixing tile's y index (reversing order)
 */
private int fixYCoordinate(int y, int zoom) {
    int size = 1 << zoom; // size = 2^zoom
    return size - 1 - y;
}
and callig it from getTile() method like this:
@Override
public Tile getTile(int x, int y, int zoom) {
    y = fixYCoordinate(y, zoom);
    ...
}
[Upd]
If you know exac area of your custom map, you should return NO_TILE for missing tiles from getTile(...) method.
This is how I did it:
private static final SparseArray TILE_ZOOMS = new SparseArray() {{
    put(8,  new Rect(135,  180,  135,  181 ));
    put(9,  new Rect(270,  361,  271,  363 ));
    put(10, new Rect(541,  723,  543,  726 ));
    put(11, new Rect(1082, 1447, 1086, 1452));
    put(12, new Rect(2165, 2894, 2172, 2905));
    put(13, new Rect(4330, 5789, 4345, 5810));
    put(14, new Rect(8661, 11578, 8691, 11621));
}};
@Override
public Tile getTile(int x, int y, int zoom) {
    y = fixYCoordinate(y, zoom);
    if (hasTile(x, y, zoom)) {
        byte[] image = readTileImage(x, y, zoom);
        return image == null ? null : new Tile(TILE_WIDTH, TILE_HEIGHT, image);
    } else {
        return NO_TILE;
    }
}
private boolean hasTile(int x, int y, int zoom) {
    Rect b = TILE_ZOOMS.get(zoom);
    return b == null ? false : (b.left <= x && x <= b.right && b.top <= y && y <= b.bottom);
}