Interesting, I came here to see if this issue was ever solved. It seems quite obvious that it hasn't. We reported this three months ago, when we had to replace a TracFone which was no longer supported on the cell network. The new phone wouldn't keep the app running, and would have to be constantly restarted.
We have an outstanding ticket open, although since the annual subscription was expiring soon afterwards, we never renewed, (and won't renew until the problem is fixed) since it was useless if it didn't autostart.
Now, to be fair to Kaspersky, it's just as likely that it's something Google changed in the later releases of Android, since Google has a bad habit of locking out useful and needed configurability, in a blatant hostility against the users (just like Apple is with iOS).