First of all, there is no such thing called Rich Trait. And triggering the Rival Questline is entirely depends on the progress of Main Storyline, regardless of how much money you have at that point. 

If you already have the Questline appeared on your log, that means you had met Tristan ( the self-proclaimed Rival ) at the Mall for the first time before. However, the progress of that Questline depends on choosing the correct dialogues for each time you meet him ( Mall > Beach > Downtown Street ). 

If you happened to choose the correct answers in your first meeting with Tristan, wait until next Wednesday and go to the Beach to meet him the second time. Make sure to choose the correct dialogues or you can retry again by next week or exiting to Main Menu. 

If you want to know if you actually choosing the correct sequence of dialogues for each meeting, basically read on how the girls beside Tristan react to your response. If they ridicule Tristan instead of you, then it's the correct dialogue and in the end they will say " Rare Tristan L ( Mall ) & Common Tristan L ( Beach ) ", indicating they keep on looking down on Tristan. 

Lastly, wait until next Thursday and go to the Downtown Street to meet Tristan for the last time. Encourage him and you'll get Heavenly Husband Trait activated.