Combine the transparency of a PNG with the compression of a JPEG.
Based on the idea from Using SVG to Shrink Your PNGs, but adapted to use data URIs instead of external images. Include on your page as inline SVG, using an <img/> tag, or as a background-image.
This technique is tested & working in all modern browsers, IE9+ and Safari 9+. Run the image test yourself to see if it works in required browsers. Inline seems to be the best option for compatibility, in which case you should use external assets so that they can be cached.
Need more SVG? Try Pixels.svg, HalfTone.svg and LinePrint.svg.
Adjust your settings before dragging & dropping a transparent PNG.
If you need to change the settings, make your changes then add the file again.
Photographic PNGs with transparency can be rather large, so processing may take a moment.