Other phrases or words that are synonymous with ‘to miss out on (something)’?

Solution 1:

If he thinks his dog jumping the fence made him miss out on the rest of the conversation, and he felt that the dog had taken away his opportunity to listen in.

then you might use : rob of.

But, you'll probably have to rephrase the whole sentence(as it only works here passively) in order to use it, or explain it better with another sentence.

Something along the lines of :

Henry wanted to know what he’d (been robbed of) by his dog jumping the fence.

rob someone of something

To deprive someone of something, not necessarily by theft. What you have done has robbed me of my dignity! If you do that, you will rob yourself of your future.

https://idioms.thefreedictionary.com/rob+of

Solution 2:

Not a single word, but "lose out" might work -

Lose out

: be deprived of an opportunity; be disadvantaged.

"youngsters who were losing out on regular schooling"

In your example -

"Henry wanted to know what he’d lose out on when his dog jumped the fence."

OR

You could just simply use "missed" -

"Henry wanted to know what he’d missed when his dog jumped the fence."