Huh. So it's immoral to get a divorce, but apparently okay to cheat, lie, abandon your wife, attempt to shaft her financially, and ignore your best friends until you need something from them. Interesting.
OP, I don't think there's any right or wrong answer here. Normally etiquette dictates that you should RSVP to an invitation, but in these circumstances, I don't think you'd be rude in simply ignoring the texts from both Tammy and Bob.
Equally however, you'd be fine in responding (either yes or no, as the case may be).
Personally, if I was in your DH's shoes, and holding out hope that a friendship with Bob could be rekindled, I'd do two things:
1) Text Tammy back saying something like "Thanks for the invite, but we have a prior engagement that date, and won't be able to make it."
2) Call Bob (not text) and say something like "Mate, you cut me off for six months, and now you send me a text inviting me for a bike ride as if everything's normal. What gives?"
Bob's response would then dictate whether I continued a friendship with him. If he said "Yeah, I know I've behaved badly, but I miss you and want to catch up" then I'd be more inclined keep the friendship. If however, Bob got defensive and didn't show any remorse or apologise, then I'd be inclined to dump him as a friend.