I'm trying to understand the question on multiple levels, so please forgive me if I seem to be rambling.
Yesterday during the rain my dog (Labrador) started barking because it's time for his evening walk, but it was raining and I was not able to take him to walk until the rain decreases.
It doesn't take a dog long to go to the bathroom (at least, not to urinate.)
Your dog did what most dog owners would expect and respect: he told you he had to go to the bathroom and/or he was eager to go out. If he has to go to the bathroom and you're not letting him go, I think you might ask yourself if this is beneficial to anyone/anything. There are options on what you can you teach your dog here.
- to stop indicating to you he needs to go out (this wouldn't solve anything)
- to dissociate going to the bathroom (less fun) from his walk (more fun): this would help you pick your time to walk)
- to stop barking to go out (this is doable and fine). To do that, he has to learn several things the usual way, with training and rewards: what "bark" means; what "no bark" means; some other way to alert you when he needs to go to the bathroom (some people put a string of bells on the doorknob).
- to tell you when he just wants to have fun, which can happen indoors or out.
Now your dog can ask directly do go out. If he wants to play, you can play with him, or if he wants to go out, well, you need to take him out. You taught him not to bark, you taught him that playing was ok inside, but you want to respect his telling you he needs to go out. Take him on a leash and let him go to the bathroom. If you don't want him to associate going to the bathroom with a walk as well, bring him inside right after, and go for your walk when it's more convenient to you. If he trusts you to take him out for a walk, he probably won't ask you unnecessarily.
This looks like a start of rainy season to me and this possibility might come again and again.
Dogs, like people, need to go rain or shine. Regarding the rain: he's a retriever. He doesn't mind getting soaking wet, shaking himself off, and doing it all over again. As far as you enjoying going out in the rain, I'll defer to @ndrewbuilder here.
I have no set schedule (I work days, nights, or middle shift, or not at all), so my dogs need to be very flexible, and they are. But they also need to trust me that I will do what they want and need sometime or another, so we get along really well.