I tried to make a subclass of Artificer that is similar to Battle Smith but uses Carpenter's Tools instead of Smith's Tools. But I don't understand how to do it or if it is even possible without making my subclass from a scratch. 

When I copy battlesmith subclass in the editor, it shows a Modifier "Tool Proficiency" feature, which states that there is a choice of 1 tool proficiency. But when I use this subclass in my character editor it just shows a text without any choice and also my character gets Smith's Tools proficiency. Where is this tool set so I can change it?

Try changing the proficiency given by "tool proficiency."

There is no tool set though. The Modifier Subtype is "Choose an Artisian's Tool". But there is no choice in character builder

Change "Choose an Artisian's tool" to "Carpenter's Tools"

the "Choose an Artisian's tool" subtype will prompt the user to select an artisan tool in the character builder.

Hello! I believe that when you open up to Edit the Choose a Tool modifier in the homebrew, I believe there should be a section that includes a specific tool listed, which is the default choice assumed. Right now you're seeing this pulled from the subclass you copied. If you change this to your desired tool, it should apply that proficiency. If the character already has that proficiency, it will instead allow them to choose another tool.

Hope that helps!

I did not do anything and it just fixed all by itself. :)