Can different CPU on logic boards from the same overall A1369 MacBook Air work?
There are 4 different CPU that fit that model A1369 according to iFixit:


http://www.ifixit.com/Info/ID-your-Mac/688


If you look at the battery removal step below for the two models you mention, you'll see the left side of the logic boards are physically different. Even in this remarkably similar case (everything else looks very, very close) I wouldn't spend more money than I was willing to lost on the second logic board.


http://www.ifixit.com/Guide/MacBook+Air+13-Inch+Late+2010+Logic+Board+Replacement/4511/2#s20602
http://www.ifixit.com/Guide/MacBook+Air+13-Inch+Mid+2011+Logic+Board+Replacement/6368/3#s27426


You surely can find a replacement board for the Core 2 Duo logic board that will fit (and hopefully it will also be reasonable cost) so I'd say you might be best putting a 2010 logic board in with a 2010 Air and not mixing model years if you needed to be sure.