Your voice is at an interesting point to grow from as you clearly have the range and you just need to kind of organize it better so you nail the pitches. I kind of came from the opposite background so what I say may be very helpful to you or not at all I'm not sure as I've never been exactly in the stage you're at. But here's what I think you should work on to help you to improve this:

first of all, on your scales:
-if at all possible, try to stop just running through an exercise CD if you do that - instead figure it out on a piano, play the notes and sing back a capella so you can repeat things to get pitches right. If your piano knowledge isn't up to that yet and you need to practice with the CD, create a setup where it's easy to pause and rewind whenever you want.
-go through them once lightly and uncompressed all the way up and down and very slowly but focusing mostly on pitch. repeat each scale as many times as you need to get every pitch right before modulating up a half step. if you can connect registers at this light volume too, great, but view that as a second priority, pitch the first priority. don't repeat scales to fix intensity issues, only to fix pitch.
-go through them again with the exact same approach EXCEPT at the same intensity you'd sing this song and only on easier scale patterns like 5-tone or octave scales
-make sure are also doing staccato scales somewhere in your workout. if you focus on pitch with them rather they will help you nail quick pitches like you have in this song. remember it's okay to compromise technique/full voice etc. a bit in favor of pitch accuracy. Btw this prioritization is also continues over into singing also - almost always. At least to the point where the pitches are close enough to not be off a jarring amount.

for song work:
-find the ranges/coordinations where pitches are easiest to nail (usually comfortable full voice and falsetto ranges) and find one or two other songs that live there, to help you practice really nailing pitch there in case it's an ear training thing. If there are no problems there then it's a high full voice technique issues so you should move on from there.
-try working this song shifted down many keys and modulate up half step by half step (like a scale) once that pitches are working in each key, and at the same time, work it shifted up many keys but in head voice, and gradually modulate it down the same once the pitches are working up there and as you do that, let the weight come back in when it wants. So basically you're trying to bring the pitch control benefits of the stable bottom and released top areas of the voice into the passaggio range. if you notice one approach, bottom-up only or top-only, is solving the issue better, you can focus just on that
-practice each phrase of the song individually and reference the original version. play it, sing it back a capella and since you are getting pitch inconsistency, REPEAT this until the pitches seem to be coming out perfectly on a couple attempts in a row.
-practice this song HALF SPEED (or as slow as you can manage) and take the time to fix every pitch as you hold it out
-practice the song on a vowel, just tapping every note staccato and try to get those pitches to land right.
-practice the song on a vowel, going through each phrase completely legato and sliding every note into the next. This will help you get better at feeling out a smooth path all the pitches are lined up on, so you always know where they will be.

As you do each of these things and get comfortable with them, try to start connecting them into each other, find new ways of exercises and song work that work a couple concepts at the same time, and don't forget to just keep singing the song through as is as well.

Have fun, this sounds like a great challenge song to take on. There are a lot of issues in it at the moment but I have full confidence you can get through them if you keep training. You are ahead of many singers in that you have all the full voice range you need available, but behind many in that you don't yet have accurate and consistent pitch control of it, so that should be the main thing you want to focus on now.