You should use the uncompressed SQL format for export. If the file is too big, there’re options to export individual tables. Select the DB you want to export before picking the Export tab. The file will then have no database name that’s meaningful.

When you import the file on the new server, first select your WP DB before picking the Import tab. Then the file will be imported into the right DB since the import file has no DB name.

With uncompressed SQL, you can edit out undesirable portions as long as you maintain proper SQL syntax. The file is plain text, use a text editor just like you might for code files. Always keep an original copy if you intend to edit the file.