Actually the choice is up to you. Many trailer manufacturers use bias-ply tires today. Bias-ply tires are an older technology and have a greater load carrying capacity than a radial.
Radials won't "follow" road scars and ruts like a bias-ply and offer a smoother "ride".
Radials became the standard design tire for automobiles in 1973 and are a litle bit heavier than a bias-ply. Radials are more the accepted tire because of the wide choice of designs and lower profiles.
Tires are part of the suspension system and act as springs. Bouncing can be caused by too much spring, too much tire air pressure, too many cord plies (harder tire) and no shocks.
No - radials will not make your trailer bounce.