Data point: TRS vs TS is a non issue with regard to tone and/or feedback. The difference between the balanced and non-balanced signal is capability for long cable run without signal degradation, and a 3db increase in volume, a significant difference to be sure, but neither of which will affect the resultant tone nor diminish/augment feedback.
The nature of feedback is entirely one of frequency (Hz), volume, and proximity: adjust any one or a combination of these and you will create, diminish, or eliminate feedback entirely.
Sounds like you already took a shot at "proximity" as you adjusted position of this and that. Also try getting
behind the plane of the speakers, or standing farther to the side of them ...this becomes a bigger deal as volume goes up! So if you are struggling with volume (being heard at the back of the room) while getting feedback, your recourse is finding the frequency to notch. Mr.Reems suggested attenuating 120Hz; try this. It has been my personal experience with acoustics (Taylors and otherwise), that around this 120-180Hz as well as around the 500-600Hz ranges is where acoustics can tend to feedback given enough volume and proximity to a speaker. I don't know the L1 system to give you specific advice, but situations like this is where multi-band parametric EQs are worth their weight in gold. FWIW, I used to use a Presonus parametric that was small and inexpensive, and perfect for finding offending frequencies. BTW, feedback also is affected by the room so while you may find the right frequencies to notch for this particular guitar, it may change +/- a hair when in a different room ...or a non issue outdoors

Hope this helps you a bit

Also, which ES is this?
Edward