You don't have to be vague about where you bought the bike. What and how much you share is up to you. But the approach about telling her nonchalantly that you've purchased a bike and are going to pick it up, seems okay to me. If she gets angry, she would have anyway......regardless of how you told her. I doubt she'll be happy about it, b/c she'll see money being taken away from her & the kids, somehow. I think she'll accuse you of being very selfish. If she accuses you of keeping it a secret, you can explain how you thought the two of you would be separated by this time...........or not. That is up to you. The thing is, she is not your mother. You don't have to explain your actions to her. This is a fault some H's fall into when they have NGS......and when they have a demanding/overbearing W.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!