Absent the neighbor's permission, you cannot cut the rosebush under California law, whether it is touching your wall or not. While you have his oral acquiesence by his statement that it was "your responsibility", I would require his express permission in written form. I would also want his general permission in written form to access is property in order to repair the wall.
If the wall in entirely on your property, it is your sole responsibility to repair (even though it may be providing benefit to the neighbor as well). If the wall is on the property line and you did not construct it, there may be an argument for shared repair, but that might be more trouble than it is worth given your neighbor's unhelpful disposition. It would also require a survey to prove it's location.
Please let me know, if you have follow-up questions.
The information is provided for discussion and entertainment purposes only. I believe in helping people and answering their questions and providing information and solutions. To encourage and facilitate this practice, it is understood and agreed by the recipient that by opening, reading, accepting, and viewing this information, no attorney-client relationship has been discussed, agreed to or otherwise established. It is understood that by posting to this site, your information is made public and no attorney-client confidence is created or possible.
In legal matters, time is of the essence. As such, you should contact an experienced lawyer right away to protect your legal rights!
I appreciate your "Accepting" my answer if it has been helpful to you. Doing so allows me to continue helping in this manner. Thank you.