Umm... this won't really work. I mean you can add some of the grit that the TS adds, and that may help some of it that is what you want.
What I use a TS for is to boost the front end of my amp. This gives it more distortion but only of certain mid-range frequencies. So I have the "Drive" all the way off. I might have it at 9 O'Clock max for some amps where I want more fizz. Then the Volume typically all the way up, but less than 3 O'Clock, then Tone to taste. Think of it as turning up your volume knob to the point of clipping.
So if you were to do:
Guitar->TSS->HiZ
On an interface, you would overdrive the HiZ input into Clipping. Which would be harsh and sound like crap. You are purposely clipping the input which is typically bad for recording interfaces, you want to avoid clipping at all costs unless you know what you are doing (ie. mastering with high end converters).
But for this application it is a no-no.
With a tube amp, or analog electronics in general, distortion is coming from clipping. Tube amps are designed to clip, which is what causes distortion, well at least designed to clip in a way we like to hear.
This, IMO, is a big reason why modeling and such is a huge challenge. Models can be really accurate if you keep the component in their specified range. But in real life analog land we are actually abusing the components and pushing them way way way out of their specified range.
Personally I wouldn't sweat it too much, it isn't the TubeScreamer that is make or breaking the tones guys are getting around here. It helps, but if they didn't have it, I bet their tones would still be quite crushing.