I have some thoughts and tips, take what you will.
Most of the "sediment" is too fine to be removed by a paint strainer.
When transferring the beer into the bottling bucket and into the bottles, the goal is to introduce
as little oxygen as possible.
Straining is a great way to introduce oxygen.
Too much oxygen will cause oxidation. Bad for flavor.
If the beer is not clear enough for your personal preference, think about giving it more time, cold crashing, and/or using fining agents like gelatin. Even if it's not clear when it's bottled, it certainly may drop clear with additional time and refrigeration after bottling and carbonating.
There's nothing wrong with drinking cloudy beer. The appearance doesn't affect flavor.
You shouldn't be transferring any "sediment" into the bottling bucket.
If you had to move the fermenter, you probably need to let it sit for some time (up to a day) before doing the transfer.
Rack very carefully to avoid disturbing or picking up sediment. If your fermenter has a spigot, make sure clear beer comes out before transferring.
Cheers