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Double Bass Can soundpost position affect bass's timbre?

The inside part of the endpin is only clamped at one side and can freely move at the other. Think about a ruler that you fix at the table at one end and pluck at the other. It resonates.
If the clamped part vibrates, it can take vibrational energy at its resonance frequencies from the body (clamped part).

The outer part of the endpin is clamped on the body and the floor, so the resonance frequencies are much higher and more damped. Think of a ruler that is clamped at both ends at two tables and plucked in between. It is similar to a string that is clamped at nut and bridge (by the force resulting by the string angle at both ends).

By cutting the endpin the resonance frequencies of the shorter inner part rise (think putting less of the ruler off the table) and the damping rises too, so less energy is robbed from the body and only from higher frequencies (not the fundamentals). The resonance peak gets lower and wider that way (less Q-factor) which has an overall influence on sound but not a strong influence on a few notes.
 
I'd also have a look at that tailwire... I can't tell what material it is from the pic, but it looks pretty big. I've found that relationship to be important when adjusting tone and feel as well.
BG
Zooming into the picture it seems that it is a spiral steel rope that is covered with grey plastic. Might be a carbon rope, but I doubt it.
At least it should be elastic enough to let the tailpiece move around with the string vibration.
 
“you've got a mass of steel inside the bass sucking up sound waves”

@LouisF can you explain that?

When I got this bass, the length of the endpin inside the bass was vibrating audibly with certain notes. I cut most of that length off, problem solved.

Whether or not you can hear it, any such vibration is "sucking up sound waves."
 
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I agree (not that it matters) with the voices saying, "go south, young man!" I can't tell where it is, East/West, But it does seem to be too far North. And, in my limited experience, that would contribute to a very bright instrument, especially if the instrument already tends that way. It is a cheap experiment and completely reversible. If it were in my shop, that would be my first inclination.

There is a reason the soundpost is referred to (in several languages) as the "soul" of the instrument: you can change soooo much in the way of sound output, quality and "feel," with a good adjustment
 
I have a Norman and Peter too. My Peter had just the opposite problem as yours. It was just too boomy and lacked definition. I could never find strings that would fix the problem and, like you, I collected a lot. I finally cut the steel endpin so only about 6" was inside the bass when fully extended to playing position and the change in tone was dramatic... much more defined and clearer. The boom was gone. I was going to sell the bass but this simple change has kept it in the family, probably for good. I don't know if it will help your shrill arco problem but it shouldn't hurt anything to shorten it.
 
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I have a Norman and Peter too. My Peter had just the opposite problem as yours. It was just too boomy and lacked definition. I could never find strings that would fix the problem and, like you, I collected a lot. I finally cut the steel endpin so only about 6" was inside the bass when fully extended to playing position and the change in tone was dramatic... much more defined and clearer. The boom was gone. I was going to sell the bass but this simple change has kept it in the family, probably for good. I don't know if it will help your shrill arco problem but it shouldn't hurt anything to shorten it.

In one of the posts above, I stated that I had previously shortened the endpin length inside the bass.