Capitalizing First Letter of Word in Continuing Dialogue

Different publishers are likely to handle the punctuation differently. I doubt that you'll be able to please them all, whichever convention you adopt.

At this point, I'll express my own preferences plus my reasons for those preferences.

Where you wrote

"We can't know for sure," he mused, leaning back in his chair, "Not until we ask more questions."

I would turn the dialogue into two separate sentences:

"We can't know for sure," he mused, leaning back in his chair. "Not until we ask more questions."

This is particularly justified here, because if you cut out the description and present the dialogue on its own,

"We can't know for sure, not until we ask more questions."

it becomes apparent that what you wrote is a run-on sentence (i.e. its clauses are grammatically entirely independent).

With your other query sentence,

I have to get out of here, he thought, But there's not enough time.

the interior monologue is not a run-on:

I have to get out of here, but there's not enough time.

Accordingly, I would put but in lower case:

I have to get out of here, he thought, but there's not enough time.

In that sentence, you don't need to capitalize but to signal the resumption of the monologue: the italics are already doing that.

I think Erik's answer is good. I wouldn't myself want to write a capital letter in either position in your examples. I'd look for some way to start a new sentence or drop the interjected phrases ("he thought" in particular).

It's not easy to see the wood for the trees when you're self-editing. Have you considered paying for an editor? A successful indie author I know, who writes for the US market though himself Swiss-German, has an American editor go through his mss. It might sound expensive, but a second pair of eyes is really invaluable. Perhaps you can find someone to do a Quid pro quo with.