Reddit mentions of Nova 48202 G3 Wood Turning Chuck Insert Type

Second/thirding the advice to ditch that and get a chuck made for woodturning, you're gonna give yourself nothing but trouble and UFO's (unexpected flying objects) with that.

They make these adapters that will allow you to use any chuck with the much more common 1" × 8 tpi threads on your Shopsmith, and if you get one of the insert-type Novas, you can can actually get a chuck insert to fit. Here is the chuck body that would go with that insert.

Another vote for the G3. You can get the insert type and use it on any lathe by simply swapping the insert to whatever thread count you need. I THINK you need the 1 1/4" 8TPI but you'll need to confirm that (could be 1" 8TPI, but I don't think it is).

Your spindle is not 1.5", probably or even 1". I'd bet good money that your spindle is 3/4" diameter and 16 threads per inch. I have a very large lathe and my spindle is only 1.25". Look for a chuck with a 3/4" x 16 thread or one that can take a 3/4" x 16 threaded insert. The standard recommendation around here is the Nova G3. It's a little over your budget but you won't ever regret the purchase. Do NOT buy the chuck you linked to. It's not meant to hold wood and the jaws are independent which means you have to adjust each jaw separately - good luck getting things centered.

Assuming the lathe is 3/4 x 16 you want
https://www.amazon.com/NOVA-48202-Turning-Chuck-Insert/dp/B0064JJ52U and https://www.amazon.com/gp/aw/d/B0064JJ88Q/ref=mp_s_a_1_2?ie=UTF8&qid=1500901907&sr=8-2&pi=AC_SX236_SY340_QL65&keywords=nova+insert&dpPl=1&dpID=41QjeodDeLL&ref=plSrch

I'm not OP, but I have a Nova G3 chuck that I like.

http://www.amazon.com/gp/aw/d/B0064JJ52U/ref=mp_s_a_1_1?qid=1452173549&sr=8-1&pi=SX200_QL40&keywords=nova+g3+chuck&dpPl=1&dpID=41WfCKmP2IL&ref=plSrch