"that" is optional here:
1. I hope that you´ll pay the bill that I haven't paid.
Note, 'that' is option if the clause it heads has a subject; 'that' is not optional if it functions as the subject:
EX: Here's the guy that I told you about.
<"I" is the subject; "that" is optional>
EX: Here's the guy that has the key.
<"that" functions as the subject of the verb "has"; it's not optional.>
"having" works, and it doesn't function as a verb here:
2. I have been enjoying having you locked in your room.