Hi Jonathan,
Interesting behavior! It almost seems like he has learned to wait for your wife's permission to go outside or he has become too dependent on the push. Something similar happens with dogs who are trained to sit by pushing on their backs rather than letting them perform the behavior on their own. This method of training has been abandoned for various reasons and a main one is because it is challenging and time consuming to get the dog to eventually sit then without the push.
Here are some ideas that come to mind:
-Just to play it safe, make sure there are no physical issues that may have caused some reluctance to go through the flap. Sometimes dogs may have achy joints or they may develop visual problems and these can interfere with their ability to hop outside the door.
-Try to troubleshoot whether there may be a fear of the flap. It could be he is a little bit hesitant of going through the flap, but is then fine when past it, so he just needs that "push" or treat incentive to get past that part. So it may help to tape the flap up and see whether the flap may be part of the problem. If that's the case, let me know and we can troubleshoot this issue further.
-Hide goodies around the yard. We can start placing super smelly treats scattered around the yard (initially right by the doggy door outside). Place these treats around when your dog doesn't see you, it has to be sort of a surprise. Then tape the flap so it stays open. Hopefully your dog will smell the treats and will hop out of the door on his own rather than waiting you to toss one.
Once out, he should start eating the treats by the door and then he may find some other treats scattered elsewhere. This treasure hunt game should make being in the yard rewarding. Make it a habit for the first week to leave several goodies hidden (or even new toys) around the yard so that your dog looks forward to going out.
Then, we can start gradually lowering the flap and start placing goodies every now and then instead of any times during the day. Making dogs find treats sometimes yes and sometimes no makes the act of searching for them quite addicting, sort of like playing slot machines in Vegas. The biggest issue though with this method is that we many end with a dog who will want to go out often just for the sake of searching goodies and not strictly to do business. Sometimes, as they head out to search for goodies, they forget about going potty only to pee or poop once inside. So this is an important potential drawback to consider. The good news is that treasure hunt games are fun for dogs and they help keep them mentally stimulated.
However, we also need to think of the big picture. With fall and then winter close, we may be back to square one with you dog not wanting to go out. If we keep on hiding treasures around the yard every now and then, can that be enough to surpass the unpleasant feeling of the brisker air or cold sensation of snow on the feet? This depends on how aversive that may be perceived and how rewarding the treats are, and over all how these balance themselves.
There are other things we can try to do to ameliorate the situation such as:
Trying dog booties
Letting your dog wear warm dog sweaters
Shoveling a path to his potty area
Make playing in the snow and rain fun: play games, toss treats, and feed your dog tasty treats
Just thought to provide a few ideas,
I hope this helps,
Best,
Adrienne