There is really no point in jumping through extra update, if target is 8.2. What is important, old head should be updated to the same version as is installed on new head. I usually always use netboot to install specific release on new controller.
I would strongly recommend professional services for this.
Headswaps are routine procedures but there is plenty of opportunity for things to go sideways.
At the very least though, you'll need to prep the 3240. Update to 8.1.4P9, then go to 8.2.3P5, including firmware, bios, sp, etc. Make sure your config is clean, the running config matches the on-disk config, and you've remediated any issues flagged by config advisor. Request 8.2 keys for the old head, just in case, and review the big controller upgrade doc. Even that doc doesn't cover every scenario, but if your config is simple and everything lives in vfiler0 your config may be covered.