Will a 3 inch (8cm) separation between CAT6 cable and 120V HV cable cause problems for 10-Gigabit Ethernet?
If you look to the NEC, which in the US is probably the only legal standard the install will be held to, you don't really have an issue.  The NEC doesn't require any separation between the NM and the Cat 6 cable.  (See 800.133(A)(2):  


  Other Applications. Communications circuits must maintain 2 in.
  of separation from electric power or Class 1 circuit conductors.
  Exception  1: Separation isn’t required if electric power or Class 1 circuit conductors are in a raceway or in metal-sheathed, metal-clad, nonmetallic-sheathed, or underground feeder cables, or communications
  cables are in a raceway 


Note that the two inches of separation applies to bare conductors - not your metallic cable!  This article applies more to routing cables in enclosures where you have both power wires and communications cables.  Does not apply to your white NM sheathed cable.  

But the NEC is mostly concerned with safety, not performance, certainly not 10GBE.  

In my experience / opinion - which is just a third opinion to add to yours and your electrician's - I doubt you'll have a problem with this, usually interference is an issue with long parallel runs at significant current.  Your 15' running 3" away along a residential 15A circuit is very very unlikely to ever cause a problem.  

If however for peace of mind you'd rather re-route that power cable, by all means spend a few bucks and have it moved.  Every mysterious network problem you have, you'll be second guessing it.  Doesn't look like it could be a very big job.  

BTW - Cat6 is available shielded, but I think what you have there is unshielded.