Here's my opinion Matt - for what it's worth....
The trem WILL affect the tone. It'll give it some springyness, some vibrato and almost an almost reverb-like effect. That would explain the Fender-ness to some extent. Also, the SC Trem has a thinner body than the SC245 (as you're well aware!) and this will 'lighten' the tone somewhat. The SCT also has the 25" scale length too. All in all, there are several factors that make the SCT more Fender-like than the SC245.
You've got PRS #6 pickups in the SCT which are clear and powerful pickups. What you may want to try is some thicker sounding pickups that will fatten the tone a bit. I think Holydivers would be great for this.
If you want to meet up and try my McCarty with HDs through your amp just let me know :)