I've spent the last 3 days trying to solve this problem and everything has failed, so i've given up on trying to figure out a solution on my own.
I'm making a 250mm extended spindle that screws onto the end of a tile saw motor. This probably wasn't the best way to go about the problem but i've been trying to build this thing in a way that doesn't depend on a lathe (even if I was using one for convenience), but with the huge spindle wobbling the whole thing vibrates too much.

There's a shoulder on the motor that's accurate enough that i'm trying to register the spindle on to, but I figure my problem is that my 4 jaw chuck is too wobbly. I was reading a 0.10mm wobble over 90mm of the bar.
Here's the offending chuck.

The bar diameter is too large to fit on the smaller 3 jaw that came with the lathe.
So far I've tried shimming the part in the chuck (i don't have any shim material thin enough), grinding the chuck jaws (I did it very carefully and very slowly, marking the jaws with a pen and feeding so fine that i'd only see about 1 spark per pass, which then increased that wobble to 0.12mm wobble over 90mm), machining an aluminium 'collet' in the 4 jaw chuck (which I really thought would work, but the bar wobbled more still).
I have checked with other ground bars so I don't think the bar i'm using is bent.
I'm really stumped so any suggestions would be appreciated.