How to optimize the order of ingredients for crêpes batter?

You really don’t want a “paste”. What you’re seeing is gluten formation. In combining flour with a small amount of liquid and repeatedly stirring, you’re essentially making bread dough and then watering it down.

Combine and mix the dry ingredients. Combine and mix the wet ingredients. Add the wet ingredients to the dry ingredients and mix well.

I have tried methods like yours and dislike them very much, since it does result in lumps, as you said.

My own order is:

• break the eggs into a separate small bowl and stir them with an egg-quirler or a small fork. Don't use a whisk or an electric mixer, they shouldn't be foamy.
• In the crepe vessel (I use a beaker so I can pour the batter directly), mix all the flour and milk at once. With good mixing technique, you should get no lumps here, but if it happens, a handheld mixer or a blender stick will break them up.
• Only now add the eggs to the batter, and stir.

I know this is a rather unorthodox order, but it works for me much better than when the eggs and milk are combined before the flour is added. And if you happen to make crepes with a flour other than wheat flour, it is even easier to avoid lumps.

Since you are using more ingredients, your additions will have to be:

• mix the baking powder into the flour before adding it to the milk.
• dissolve the sugar in the milk before mixing it with the flour.
• add the melted butter and vanilla at the end, after you have added the eggs.

The appearance won't always be perfectly smooth, because you use fat in the batter, but you shouldn't have actual lumps, just some unevenly dispersed fat droplets, if anything.

As you noted in a comment, this certainly requires that you follow a recipe and measure out all ingredients, instead of adjusting the consistency as you go. Luckily, this has other advantages beyond getting rid of lumps, so it is worth the switch.

Something I noticed when looking at your pictures a second time and reading the "they melt" comment - it seems that your melted butter is not especially warm, and your batter is cold, and the butter just resolidifies into these lumps. So even with my method, I would suggest to use warm ingredients and maybe switch to using oil instead of butter. Room temperature eggs will give you a slightly better end result anyway, and if you warm the milk before mixing, it will help dissolve the sugar. On the happy side, as you noticed, butter lumps are not much of a functional problem, as opposed to flour lumps - they are just ugly, and that at a stage which only you get to see.

When I do crêpes I put first the flour, a pinch of salt, I dig a little well in it (un petit puis), I add the eggs and oil, start whisking, and then add milk little by little, constantly whisking, and that makes a perfect consistency! And it's much more simple. And don't forget a little bit of beer to make them even lighter!