When the service is not responding for too long, the app uses a build-in ringtone. When alarm starts and your phone is locked we are unable to change the playlist or album in your Spotify app. In this case the app will play simply where you last left it.
I met the same issue and figure it out by downloading Spotify playlist via the AudFree Spotify downloader. Then set the offline Spotify files as ringtone. Hope it can help you.
Really ? I heard AudFree before,but I haven’t use it,I think I will have a try.did you know another software Tunelf Spotify downloader? I will consider to use too. 
If you want to set Spotify as an alarm, one ultimate way is to download Spotify as MP3 local file and then transfer it to your phone. You can change the soundtrack any time as you like. Moreover, by doing so, you can listen to the Spotify songs on any device or player without limit.
I got the same issue and figured it out by a workaround. I download Spotify to MP3 by using AudKit Spotify Music Downloader. Then listen to the Spotify local MP3 files offline without opening the Spotify app. Fixed!
Yes, I have met the same issue that alarms don’t play the selected ringtone I choose from Spotify but a classic ringtone on my phone. It seems that you said right as my phone is locked.
Based on this, I don’t select a ringtone from my Spotify app but use Spotify Song Downloader to download Spotify songs as local files and then choose a ringtone from my phone to set Spotify music as an alarm.
Hello, sorry for the issues. Our Spotify integration is now based on reverse-engineered intents from the Spotify app… We have found how to make Spotify play a certain playlist or album, but unfortunately this method does not work when screen is locked. In this case Spotify will simply play where left off…
So a workaround for this is to disable the lock a screen which I understand is not really convenient in many cases…
But at the moment we do not know a better way how to implementing this…
Oops! I am wondering whether it is because you forgot to confirm to set a new ringtone. Haha, everything can be possible. Or are you sure that the song you want to choose is allowed to play in ringtone settings? I mean, did the ringtone system recognize your desired song as permitted? Since only MP3, M4A, and some other audio formats are supported, check whether your track is one of those. 
If you fail to find the real reason, you can directly go to AudBite and other music converters for help. Hope this can work for you!