340 .040 Bore Okay?

Most of the time it would be ok, but core shift means it will have to be sonic checked to be 100% sure. If the block is already at .030 the machine shop may just hone it to .040 if it looks like it will clean up there.

My experience has been that overheating on a rebuild is usually more likely to be tune related than overbore.

FWIW all Blueprint engine chrysler small blocks are bored .040 as standard practice.
The website for Blueprint Engines specifies that the engine blocks are sonic tested, so they are making sure the block is capable before throwing in a .040 overbore, and that is on 360 based blocks.

I would use this as reinforcement that the block should be sonic tested and not as a free pass to bore any block .040 over stock size.