I had a sword (one) that broke in Battlegear mode (dual weapons) and when it broke and I tried to use the hand to fight, the game crashes. I have notified the makers of Battlegear of this as well. I took InventoryTweaks out and the BattleGear works fine, so it's an issue I think with InventoryTweaks trying to replace something that cannot be replaced like normal...

Video of Crash... http://youtu.be/jS2j1Ly5IXA?list=UUv8LfKG3Oj9D3vGr_0XNzCw
Crash report(s).. http://filebin.net/pulenfzk0d

There doesn't appear to be anything about Inventory Tweaks causing issues in any of those logs.

You shouldn't be modifying your Minecraft.jar however:

[16:55:23] [main/ERROR] [FML/]: There is a binary discrepency between the expected input class bpz (bpz) and the actual class. Checksum on disk is 3f6a8b55, in patch 894c3a10. Things are probably about to go very wrong. Did you put something into the jar file?
[16:55:23] [main/ERROR] [FML/]: FML is going to ignore this error, note that the patch will not be applied, and there is likely to be a malfunctioning behaviour, including not running at all

I never manually changed anything, might have been either Optifine or Lightloader would be my best guess... I know Lightloader requires you to add -Dfml.ignoreInvalidMinecraftCertificates=true -Dfml.ignorePatchDiscrepancies=true to the JVM arguments for it to run