the post, bridge and bass bar all work together with the post acting like a fulcrum, the bass bar being a spring (and in a sense like the end of a lever) and the bridge putting it all in motion. Accepting the given that the E string will require the most movement of the top, and of the bass bar, lengthening the 'lever' will facilitate that. Shortening the 'lever' will hinder that.
Very much agreed. Well, for horizontal (or rather: lateral) string vibrations, as from arco, it's the strings putting it in motion, and the bridge is the lever. The conventional proper DB jazz pizz also has a significant lateral component, combined with a vertical component (which is less for arco). There are 2 ways to change the length of the lever: string height & SP position.
Increasing the E height was on the list, but assuming this pertains to the OP's 1870 Thibouville Lamy bass, there is quite a narrow band of heights that are acceptable; the E-height being restricted by bowability of the A & the fact that the neck angle is fixed.
The other way to increase the length of the lever, for particularly the E-string, is to move the SP towards the G-side, as you say. Following the model of the lever, that should increase the E response, and suggesting that was my initial reaction. But personally I have little experience in moving east-west; very quickly it requires making a new post. There once was a bass for which I made several SPs, but I can't quite remember the results... remember I am not a pro.
So I looked it up, and the common advice (for a violin) is summarised by this:
which is exactly what Chet suggested for this issue: towards the E-side.
This may work because the post is not stationary, but also moves & actuates the back. And if you move the post position on the back inwards too, the location that it drives moves away from the ribs & becomes 'softer'. So you get more motion in the back.
Particularly for pizz, the other thing to consider is the vertical string motion (up-down). Pizz has a stronger vertical component than arco. For the vertical component it is clearer: moving towards the G-side makes the E-side of the top softer & more responsive (again by increasing the length of the lever). However, the actuation on the back becomes harder, with the post moving towards the rib. The net result depends on the instrument.
Overall I think the result of moving the post either way depends on the specific instrument, as well on what you want to optimise (pizz or arco). So trial & error, as the luthier has probably done by now. And it may well be that for basses, with a higher breakover angle than violins, moving east is generally more effective than moving west for this issue. Or maybe set the post at a slight angle?
Too much tension on the top can choke it so heavier strings might make it worse.
Yes, agreed. This I tested extensively in my test bass, wherein I can change both neck angle & saddle height (up to 9.5 cm above the top), and on that instrument lower load always gave stronger lows & fuller sound (at the same bridge height). But an instrument like the OP's, if it is what I think it is, should be able to carry mature strings, at least for a trial.