some money in an account that he'd set aside

Yes it is optional. In formal writing (like an academic paper) it is better to omit it. In conversational speech, I'd say it is more natural to keep it

It is optional, although the “that” makes the sentence sound less formal.

Yes, when "that" comes before a pronoun you can omit it. If it is followed by something else, like a verb, then you need to keep it. "The dog that I saw had yellow fur." "The dog I saw had yellow fur." "The dog that barked at me had yellow fur." (incorrect) "The dog barked at me had yellow fur."