Aborted connection (Got an error reading communication packets) MySQL on Amazon RDS
Since you tried the above steps, your sticking point seems to be the one you can't alter.

I wrote a post back in August 2012 (Local database vs Amazon RDS), mentioning how the number of DB connections and the Buffer Size are fixed and immutable for each in RDS server model. I also mentioned that the Transactions Logs are fixed at 128M, no matter what RDS server model you use.

SUGGESTION: Be kind to yourself. Migrate the DB from RDS to EC2. Then, you have the necessary flexibility to change the InnoDB Infrastructure. Consequently, you will have to write your own backups and setup replication by hand.

I know you said going to EC2 is your last resort. Since you cannot tune InnoDB effectively in an RDS environment, EC2 is the next logical step.