You go to module builder and you select the import option on your custom module. Then you redeploy it so it will be ready for imports. (The problem with that is that any customization created after de initial deployment will be lost. I might affect your records as well)
The other option is to modify the backend files of your custom module. Open file \modules\YourCustomModule\YourCustomModule.php
change:
...
class YourCustomModule extends Basic
{
public $new_schema = true;
public $module_dir = 'YourCustomModule';
public $object_name = 'YourCustomModule';
public $table_name = 'YourCustomModule';
public $importable = false;
...
To:
...
class YourCustomModule extends Basic
{
public $new_schema = true;
public $module_dir = 'YourCustomModule';
public $object_name = 'YourCustomModule';
public $table_name = 'YourCustomModule';
public $importable = true; // PAY ATTENTION TO THIS LINE
...
Once you have done the change, you need to perform a Quick Repair & Rebuild and you should be able to import into your custom module.
Please try it there to see if you can reproduce the problem, and provide me with a detailed “steps to reproduce” including the exact things you set up in Module Builder, and then the exact procedure to start the import until you see the error. Thanks.