I'd like it to skip to the next song without starting playing it
See, I don't really understand what this means, because they seem mutually exclusive.
Skipping to the next song without playing it isn't skipping to the next song. It's just stopping.
Having stopped, if you later didn't want playback to restart from where it was before, click the Next button instead to start playback from the next song.
This sounds like the same result to me, but I've never used iTunes to compare it to, so maybe I'm missing some obvious scenario where the difference matters.
Someone with iTunes knowledge might be able to suggest a better alternative, though it's possible the differences in how MusicBee and iTunes handles play queues means there is no equivalent.
Bee excellent to each other...