Nice to have you here. Welcome!
Your partner needs to be treated because, not only is he probably giving chlamydia back to you (after you've been treated), but he's giving it to his other partner as well. All three of you need to be treated at the same time for this bacteria to get out of your lives.
Have you thought long and hard about this relationship and the risks associated with it? You're really dealing with a huge unknown at this point. His other partner could be sleeping around which, utimately, could be putting you at risk for STI's - even the dangerous ones.
If walking away is not an option, please consider condoms.