Jisho

If anything it would be 歌わさせられる, but this is a a godan verb so it goes with -ぁせられる and becomes 歌わせられる.

-される ist the causative ending for ichidan verbs. For godan verbs the causative would be -ぁせる making it 歌わせる.

the inflection shows 歌わせられる which is Causative passive full form, consisting of 歌わせ(る)& られる (because 歌わせる is no longer godan verb --> use rareru as it is passive form for ichidan verb.

歌わされる is a shortened version, which comes from 歌わす (shortened causative form of 歌う). 歌わす is godan verb instead of ichidan verb as the non-shortened one is ==> the passive form of 歌わす is 歌わされる as you said.

Final:
歌う (dictionary form) --> 歌わせる (causative form) --> 歌わせられる (causative passive full form)
歌う (dictionary form) --> 歌わせる (causative form) --> 歌わす (shortened causative form) --> 歌わされる (shortened causative passive form)

the inflection shows 歌わせられる which is Causative passive full form, consisting of 歌わせ(る)& られる (because 歌わせる is no longer godan verb --> use rareru as it is passive form for ichidan verb.

歌わされる is a shortened version, which comes from 歌わす (shortened causative form of 歌う). 歌わす is godan verb instead of ichidan verb as the non-shortened one is ==> the passive form of 歌わす is 歌わされる as you said.

Final:
歌う (dictionary form) --> 歌わせる (causative form) --> 歌わせられる (causative passive full form)
歌う (dictionary form) --> 歌わせる (causative form) --> 歌わす (shortened causative form) --> 歌わされる (shortened causative passive form)