Personally, I think it would be easier to manage minor gain adjustments on the DR-680 rather than the V2 given that the Tascam has the ability to gang channel together and a continuous variable gain range. With the v2, you've only got 5-10dB to work with for minor adjustments and will have to adjust the stepped gain, possibly mid-show, for larger adjustments.
IIRC, the trim on the V2 is attenuation (unlike the V3 which I believe is additional gain), so you're technically adding resistance and noise by not running the Trim at 0dB. The noise added by the trim is probably negligible, but it's noise none-the-less.