Trying to upgrade the wheelset on a Concord mixte. The issue is the original front wheel had a solid axle, while my 700c front wheel has a larger diameter quick-release axle. I don't care about having quick-release on this bike - is it simply a matter of replacing the axle and cones on the wheel? Will I be able to find a solid axle/cones to fit a newish hub?

That's hard to say. Cones differ in their profile. Substituting a different cone may cause the bearing to ride too high or low on the race. The former will decrease it's ability to handle lateral loads while the latter decreases the ability to handle radial loads. Basically, it's a trial and error proposition.

Your other option is to take a file or Dremel tool and open up the dropouts a bit.

That certainly occurred to me. I'd prefer to take a non-destructive route if reasonable, even though the bike is nothing special.

As I look at the problem, non-destructive would seem to be to build a new wheel on the hub of the old axle... or find a wheel built on one of the axles with flats milled into them. Changing out the fork is another option, though I suspect it is not one you'd favor.

Just file a couple flat spots on the axle.