I need advice from beagle parents who happen to have a cat(s).

Long story, short... I'm taking in a stray kitten. She's in the process of getting used to staying indoors. What worries me is how my beagle will get along with the cat.

The two of them have had brief interactions. I feed them at the same time, in the same room. No problem. On one occasion, I brought my beagle outside to say hello. She was cool with the cat until I let go of her collar. She chased the cat out of the yard.

Today I brought the two together for another meeting. The kitten was cool with being in the same room with my beagle. I'm not sure how my beagle felt. She laid in her crate the entire time without showing emotion. Beagle didn't chase or bark or growl. She didn't want to play and showed little interest in treats. After I let the kitten out, the beagle was back to her regular self. How should this be perceived?

Please don't redirect me to another board. I need advice from a beagle owner.

If i take the kitten to a shelter or leave her outside then it'd be a guaranteed death sentence.