what would be the primary disadvantages of splitting my tables into table spaces and/or separate schemata
DISADVANTAGE #1
InnoDB only has one Buffer Pool. The benefits of partitioning or compartmentalizing database tables for different clients can still be bottlenecked by database buffers (log buffer, InnoDB Buffer Pool)
DISADVANTAGE #2
Scaling becomes dependent on manual operation. This applies to scaling up disks, hardware (disk controllers), and memory.
DISADVANTAGE #3
More tables means more open file handles to cache distinct tables. Requires more flush to disk.
DISADVANTAGE #4
Introducing new groups of related tables regurgitates the first three(3) disadvantages and  exacerbates the overall design.
See my other posts on such setups and other drawbacks

Nov 15, 2011 : One Big Database vs. Several Smaller Ones
Mar 11, 2011 : What problems will I get creating a database per customer?

SUGGESTIONS
You might have to compensate these deficiencies with lots of RAM with the InnoDB Buffer Pool and the Log Buffer getting the lion's share.
You could create a hybrid setup where InnoDB Log Files are on one disk and InnoDB System Tablespace is on another, and all the data on a third disk. This concept is well expressed in YOSHINORI MATSUNOBU'S BLOG (FaceBook DB Engineer).