There are ducted mini split systems that may be useful to you.

Pushing (or pulling) air from an existing duct system is not a great path forward, as balancing the flow is challenging and critical. For example, if you could plumb the mini split into your circulator, what happens if they’re both on at the same time? The mini split could frost over. And if the circulator isn’t on, you could be back-flowing the air intake of the circulator.