Thank you for reaching out to Constant Contact API Developer Support.
As of right now, custom fields are not available through the bulk import process in our V3 API, however this is a feature that our development team is actively working on. I can't say for certain when this will be available, but I am hopeful that it will come with our next major release. You can see our latest release notes as we update the API here: https://v3.developer.constantcontact.com/api_guide/release_notes.html
If you are working in our older V2 API, you can currently use custom fields with some limitations. The V2 API can only see and use custom fields that follow a very specific naming convention (such as "Custom Field 1" for example) as it was designed with our older contacts system which did not have support for named custom fields. While it is possible to create custom fields with other naming formats using our website, the V2 API cannot see or interact with custom fields outside of that convention.