Hi,
First congratulations for your upcoming marital plans!
The 1200 from KGX doesn't call at Darlington, unfortunately, but the 1230 does. If your flight arrives on time then you shouldn't have any problems catching the 1230. This arrives in Darlington at 1513 and the connecting service leaves at 1530 from across the opposite southbound platform, if I remember rightly, but do check, and arrives into Middlesbrough at 1555.
If you want the cheapest fare then book well in advance (as far ahead as possible). These fares are notorious for not being available for when you want to travel. Midweek has good availablity, weekends not so.
A handy note to remember when travelling on a restricted ticket. If for whatever reason you miss the train on which you're booked (tube delays, plane problems, etc) then just go to the ticket office and explain what happened, they'll verify and stamp the back of your ticket to allow travel on the next train.
Have a great holiday!