Your new format is a totally different format although those techniques can be used for the background image. It uses a totally different binary, so they probably will not be compatible. Second of all, both compressions might be totally lossless, but when combined, destroy the image. If you have to choose one, chose the PNG.

Which compression technique is preferable, deflate or wavelet? Does deflate or wavelet preserve higher quality? Does deflate or wavelet reduce more unnecessary space? What are the problems of each techniques? (Quality is higher priority than space, but if both are equal quality, space matters.)