I mount about 2000 orchids a year so I have a lot of experience mounting plants in flower.
You can mount anytime. The danger is that if you damage the roots on a Cattleya, you may lose the flowers. I have occasionally mounted flowering Cats and been successful. Cattleya actually can be mounted on rather small branches. The plant will grow around the branch, sideways and down.
This Bassovola is so large you can not see the mount it is covered in roots (about 8 years).
Oncidium flowers get their strength from the pseudobulbs so I always mount these in bloom and seldom have a problem. Damage to the roots do not effect the flowers.
Phals can be mounted in bloom if you are careful not to break the roots. It is better to start with a Phal that is not too large. Phals grow down so it is best to mount the roots at the top and the plant and flower hanging.
I wrap the entire root ball (after removing most if not all medium) in sphagnum. This eliminates the stress of mounting bare root. The new roots will grow on the outside of the sphag and over a couple of years the roots in the sphag will be discarded by the plant. It is not necessary to ever add more sphagnum. I have done this so often and now so gentle, that my plants do not even know they have been moved.
Practice will get you there, but the worst is that you may lose a couple of flowers.