Robo Hippy has documented his "sanding in a barrel" setup here
sanding hood
A sanding booth is a good idea but you will need to plan for sufficient airflow and effective filtration (or exhaust to the outdoors). Spray booths typically are designed for 50-100 linear ft./min. If the booth opening is 8' x8' (64 sq.ft.) that would call for a blower moving 3200-6400 cfm while accounting for any loss due to filters. That's a lot of air- if it's going outside you may need a heated makeup system. If it's being recirculated you still need a big blower. It's more efficient to restrict the catch area with good hood design. You might think about designing a "booth" with wall and ceiling panels enclosing the lathe closely to keep the airflow requirements down.
Filters are another big subject. MERV 13 furnace filters are not going to capture the smallest particles. Some people insist on HEPA level filtration which is supposed to snag most particles down to .3 micron. A common rule of thumb in dust collection systems is to have a minimum1 sq.ft. of filter area for 10 cfm of airflow. That gets expensive fast. That ratio might not be applicable to a low velocity "booth" though. You also need to consider cleaning the filters- a single stage system will blind the filters with fine dust relatively fast and anyone who has had to clean out a cartridge filter after accidentally clogging it knows what a pain it can be.
I assume most turners (and woodworkers) in general) struggle with this. I have a 6" drop from a 3hp cyclone system to a rudimentary hood on a pivoting arm behind my lathe, and I run a 2,000cfm ceiling mounted ambient air filter, plus I wear a papr when sanding.