Thanks for your question.
Believe it or not it is not uncommon for some dogs to hate going out when it is raining and, I for one, cannot blame them. However, it is important that our dog goes out in all weathers and does so without stress.
Without more information I would be unable to give a more detailed/bespoke response to your query but I can give some general advice regarding this.
Basically, we need to change the relationship of the rain to your dog. Currently, it views this in a negative manner and is very reluctant t venture out and by the process of having to drag it out then this only compounds these negative feelings.
To change (counter condition) this we need to modify your dog’s perception to that of being positive. This will take a bit of time but can be done and, unfortunately, relies on it actually raining in order for this counter-conditioning training to be put into effect.
The next rainy day, after the morning walk, set up a play area quite close to the front (or back) door. With the door closed play a scatter feeding game with a very small amount of (very small) treats. Initially drop the treats nearer you than the door. After about 5 minutes or so take a break for half an hour.
After this time, assuming that it is still raining, you can resume the game in the same spot but this time throw the treats a bit further from you and a little bit nearer the door however, only throw one treat at a time and don’t release the next treat until your dog has returned to you. Again, after about 5 minutes take another break.
Repeat the above until the treats thrown are right next to the door.
The next stage is to start the game again (throwing the treats next to you initially) with a window in the same room open (so your dog can hear the sound of the rain). As before build up the training until you are throwing the treats right up to the door.
If all going well then we can up the ante again and very slightly open the door and start the game again (again initially treats nearer you than the door).
Now open the door a bit more and repeat the same gradually building towards the door.
Keep doing this until you can open the door fully. As I say, this will take a bit of time (at least a few days, if not weeks, to get to this point.
Once you are at the point where your dog is happily engaged with the game with the door fully open then you can throw a couple of treats just on the other side (outside) of the door. This time when your dog collects the treat call it back in and give it plenty of praise. Gradually increase the distance beyond the door and each time your dog returns to you after collecting the treat then give it plenty of praise.
Once we get to this stage you can clip on the leash, throw the treats, walk through the door to the (outside) treat, and, after collection give your dog plenty of praise then go straight back in and give your dog more praise. Repeat this reducing the speed of your return to the house after your dog collects the treat.
Hereafter, we move from throwing a treat to just going out the door (in the rain), giving a treat and plenty of praise when in the rain and then going back into the house. This is all done with the leash on.
From herein, it is just a matter of increasing the distance until you get to about 10 yards or so from your home. Once you get to this point you can reduce the treat/praise reward to your return only and increase the duration of being outside. Whilst outside ensure you give your dog plenty of positive interaction and game play (tug toy, fetch etc). We are looking for your dog to focus more on you, and the fun, than the rain. If this is repeated your dog will eventually link fun with rain. In addition, as you will be leaving the treats/big praise until you get home then your dog will look forward to the return as much as going out.