Has the backing plate ever mounted on the spindle correctly? , if not have you checked it with a thread gauge, hardening has 2 1/4 x 10 tpi spindles they look close. Been there with a spindle thread protector, just sayin
You can remove the backing plate from the chuck . It'll be easier to do then spinning all that on your lathe . It's only about 4-6 bolts usually. Plus you can clean the chuck repair and relubed as needed. I wouldn't mount it in the four jaw with the chuck mounted . Just my opinion.
Also, you can put the Dykem on it then work it on to where it binds. Then look at it closely to see where it is binding., But agree with Dlane as unless it has fit prior it may be a little diff thread, I know the Jet 10x24 I have is a Metric size with a 8 tpi Kinda diff and hard to find.
after you remove the backing plate try to tread it on back wards, more than likely the thread is a little under size in the chuck. You may be able to lap it in with valve lapping compound, i had on years ago that did the same thing.
Do me a favor and measure the openings etc on that backing plate. It almost sounds like it fits my 10X24 If so I would possibly like to purchase it off you. It sounds like it is the metric size X8 that fits mine. Then you could put a new one on the chuck.