Will start with most obvious. SST theoretically has active pickup couldn't find spec's with a cursory on line peek and while it's unlikely that voltage would exceed M box
max 8.7 (even 'hot' piezo's wouldn't typically run more then 5 v) but check which input you're plugged into (choice of two) . . . If what would tend to be first choice DI doesn't seem to work no matter how gain is set, try the line in
The clipping and distortion is unrelated to efficacy of tracking through the board and unlikely to be fixed by software re amping
Some models of this guitar (or productions) had QC issues with grounding the active electronics . . . Wasn't conspicuous live but posed problems when recording . . . But would not characterize those issues as 'clip/distortion' more the expected bad ground hum/buzz a typical tech can sort that out in minutes
Once initial input gain situation is addressed you might still have issues with how suited the native onboard electronics are for DI recording. I have actually used this guitar in a very similar situation but always as composition scratch track . . . Less concerned with finished tone then not disturbing neighbors. Factory setup (on board electronics) is pretty MOR and depending on what you're looking for you or trusted tech can swap out the FETs and pots and tailor the sound at bit more for your purpose. Though if you're happy with live sound you might not want to mess with that
In any case there is no realistic way you are going to duplicate your live amp 'tone' via strictly DI tracking. I have and use a variety of modelers and software re-ampers not particularly critical of any of them but it's purely a matter of personal taste as to which works for you. Unfortunately, addressing one specific product, the full priced version of Amplitube is just enough different that the entry 'bundled' version does not demonstrate accurately strengths and weaknesses . . . Tried the entry 'bundled' version was under whelmed, inherited an unregistered full price version and have found specific uses . . .
Still typically prefer to 'amp' & mic but there are numerous instances when re-amping via software is appropriate and even though, based on experience with entry product I never would have purchased it Amplitube is used as often as the other re-ampers I have
Main point in that is that one size will not suit everybody. Personally I used the Chet SST I had through an AC15 and was pretty happy with results. Had a guy sit in on what was essentially a live acoustic show and he insisted on using it through a Fender (maybe twin, can't remember for some reason . . . Wasn't the Princeton) and we both really hated the results (he blamed the guitar I though he was an idiot and would have gotten better results using my generous offer of the AC15) . . . Interestingly enough that wasn't the last time he sat in.
But I think you have two issues and clipping and distortion needs to be addressed first..
(regularly do live shows with a guy with a pair of Taylor acoustics, for one we regularly run through an external DI into board (and it works fine) the more recent model works best plugged straight into boards 'line in' option (while we've done this a number of times there just has not been 'time' to actually trouble shoot or even sort out the issue they're low rent shows, we don't live in same area and he shows up about 5 min. Before he goes on . . . So we default to what ever works.)