Two things:
One -- Setting the "Layer" on your UI objects is not going to solve this issue. As far as I can tell, all UI elements in a canvas with Render Mode set to Screen Space - Camera will output to that camera with whatever layer the parent canvas is set to.
For example, if I have a Camera X that only shows Layer X, with a canvas set to Layer X, and a UI Panel inside that camera set to Layer Y, the UI Panel will still be rendered to Camera X.
Conversely, if I have a Camera A that only shows Layer A, with a canvas set to Layer B, and a UI panel inside it set to Layer A, that panel will not be rendered to Camera X.
To solve this, you'll need to set up a "Bloom Canvas" and "No Bloom Canvas", and have Bloom Canvas render out to a Bloom UI Camera with the post processing applied, and No Bloom canvas rendering out to a No Bloom UI Camera.
There's one other trip-up you may encounter:
Two -- Currently, post-processing effects also apply to any camera lower on the stack. So for example, if I have a base camera, a bloom camera (with "post-processing" checked), and a non bloom camera, if I stack it as:
- Main Camera
- Bloom UI Camera
- No Bloom UI Camera
The bloom is applied only to layers 1 and 2. Like so:

However, if I stack it like this, with no other changes:
- Main Camera
- No Bloom UI Camera
- Bloom UI Camera
The bloom will be applied to layers 1, 2, and 3. Like so:

So when applying post processing effects, make sure you have your "non-post processing" cameras always at the top of the stack.