MailerLite API doesn’t seem to allow adding users to specific segments or campaigns. It also doesn’t look how it supposed to work anyway.
So if you have a segment for each product, you should also create a group for each product. Then you can configure each segment to only have users from a specific group.
Example:
You have Product 1 and Product 2.
You have Segment 1 (for Product 1) and Segment 2 (for Product 2).
You need to create two groups in your MailerLite account – Product 1 and Product 2.
Then set corresponding group for corresponding product on your site.
Then configure each of your segment for corresponding group. E.g. Segment 1 should use users from group Product 1.
This way it should work how you want.