Being rather stubborn, I have often completely redone setups on problematic violins - sometimes multiple times - and often the uninspiring nature of the tone was still evident. That said, I have often been amazed at the positive impacts of small adjustments. Sometimes the adjustments are simply things that I overlooked in the initial setup, e.g., string height at nut, or bridge thickness, or optimum bridge placement .
If there is a logical formula to apply, I haven't found it. I do usually start with SP tweaks and end up at string selection. Adjusting violins for optimal tone continues to be both rewarding and frustrating for me. I do spend quite a bit of time on each violin I own (which is too many) because they only sell if they sound good. Most of the frustration ensues when a violin that should sound good, based on wood, arching, thickness, doesn't, and won't respond to any amount of adjustments. Though they may not help you, I will share some of my adjustment ideas in hopes of eliciting more participation in this topic.
Regarding E string tension, the other variable to adjust is the relative height of the strings on the bridge. Although there are limits based on standard string height above FB, there is a bit of leeway to make adjustments. I have found that bridges that are relatively horizontal in profile (E string not much lower than G string) tend to produce harsh, unpleasant E/A strings and weak D/G. Lowering the height of the treble side (or making a new bridge with slightly higher bass side) seemed to help string balance.
Sound post tweaks are where I spend most of my time and I seem to go through phases where I like them closer to, then farther away from the bridge foot (even though each violin has unique "needs"). In general, I find that, in addition to the closer = tighter, further = looser/darker correlation, closer placement puts more emphasis on A and E and farther behind bridge foot seems to allow the bottom end to speak up. East/west adjustments seem to be more specific to each violin.
One frustrating aspect of SPs for me is that the post can look straight one minute, then appear crooked the next. I find that tiny N/S adjustments to the bottom of the post can have significant impacts since they can correct a post whose top surface exerted pressure unevenly - not through the center of the post, but either through the top of the post closer to or further from the bridge foot. I know SP tension has been discussed before but I hope the discussion continues as I also tend to go in phases with that.
With regards to bridge tweaks, I do know that the more time I spend on any particular bridge, the thinner the bridge and more open the cutouts become. Most of my tweaks are to the lower half of the bridge where I have been experimenting with ankle width, shape of bottom arch, thickness between bottom arch and kidneys. Most of the tweaks have produced positive results but most likely I am just moving towards better cut bridges instead of really finding out what does what.
Moving the entire bridge east/west by even 0.5mm can sometimes make dramatic improvements in tone and I don't know whether it is due to a better relationship of bridge to bar/post, or whether the nut/bridge/end pin alignment was improved. Speaking of which, I will sometimes pull the tail gut in either east/west direction to see whether that improves the tone. If it does, I will take the setup down and check that the endpin is in the right place.
This is a tough topic to articulate but hope others chime in.