Does code allow a main-lug-only panel connected to the meter lugs?
Can the main lugs in the meter can be double tapped like this and
  still be within code?


The short answer is NO. NEC 110.14 States: Terminals for more than one conductor ... Shall be so identified. I have never found a 200A residential 120/240V meter base that had an identified multi-tap fitting, and the utility companies I have worked with forbid it. 

That being said, it is allowed to have a maximum of 6 service disconnects at the meter. So it is not unusual to put a small panel beside an existing Main and add an extra breaker and leave space for additions. All that is wrong is that there must be something like a junction box added and tap the conductors there instead of at the lug. 

If you electrician balks about it you can always get the AHJ or the utility company involved and see what they have to say. At least I would contact the owner (not the electrician in the field) of the contracting company and see if they want to get them involved as a tie breaker.

Hope this helps.