The Brahms Institute in Lübeck has a photograph which it catalogues as being of Ernst (NO. 30750). In some ways it resembles Ernst (the hair-style and slim figure) but I don’t think it’s actually of him, and the hunting whip and posture are most uncharacteristic. Decide for yourself here:
Click on the home page, then go to ‘Digitale Bestände’ followed by ‘Fotographien’.