"Texture.wrapMode" is, in this case, working as intended.
The API doesn't record a command into the CommandBuffer, hence the texture's use of the last TextureWrapMode set during Start().
A new "CommandBuffer.SetTextureWrapMode" API would help in this case, as it would allow "Set TextureWrapMode to X on A" commands to be recorded into the CommandBuffer.
We have not received feedback/a feature request for something like the above before: our recommendation, in case workarounds are proving to be too inefficient, would be to provide us with feedback (See "How to check the roadmap and provide feedback": https://forum.unity.com/threads/how-to-check-roadmap-and-provide-feedback.1321341/) so that we may prioritise appropriately.