A spacer band isn't going to make your rings sit flush unless you have one custom made to fit perfectly against the engagement ring. People usually get spacer bands if the diamonds on the wedding band would sit high enough to scratch the prongs on the engagement ring. That doesn't appear to be the case with yours, but I can't tell for sure.
That’s one of Alex Park’s settings, right? We need to see a side profile shot to see if he placed a donut and if so, where. I have a few of his settings, and getting any ring to sit flush needs a side angle photo, looking across, not down.