Ted this is an FAQ of interest to many users, so I'd like to elaborate a bit on Joe's comments; perhaps you've slightly misunderstood. The ratio of interest here is the preamp's OUTPUT impedance vs. the power amp's input Z (Z = impedance). The P.A.'s input Z should be at least 10 times the pre's output Z. So if you select 100k-ohms input Z, the pre's output Z must be 10k-ohms (or less) to avoid lo or hi frequency (capacitive or inductive) frequency rolloff within your interconnects.
The BAT website does not offer spec's for the 5i, but looking at other models they do show the highest output Z of their line (the model 3i) output Z = 1k-ohm; other models are even lower (better). So your 5i's output Z is probably not the issue here.
Regarding gain, for best noise performance use the least amount of gain that you can, while still attaining full output drive capability (can you drive it into clip, or at least as loud as you desire to?). Try strapping for minimum gain & then work up from there if that's not adequate.
I am thinking that you may need to experiment with upgrade AC cords on the preamp, until you find one that gives you the tonality charactaristics that you prefer. Tube rolling, as you refer to, is of course another avenue of approach to attaining your desired charactaristics. This can also be combined with your AC cord tuning.
Finally you may need to experiment with component shelving, & or cones, & or vibrapods (isolation & vibration treatments). Like upgrade AC cords & tube rolling, this is an entire area of speciality unto itself & I am not going to attempt detailed explanations here.
Please use the forum search engine to locate the multitudes of detailed posts regarding all of the aforementioned techniques. There are many available tools at your disposal. Your choices of components above are very good; I suspect that you'll be easily able to achieve the desired results when you find which techniques work best, via experimentation, which is the only way to find out. There are no specific rules to doing this.