I assume you did not have this vibration before.
If the yoke seems to have excess play, then replacing it may help.
I don't think a crossmember is absolutely necessary. The t5 felt like it weighed less than the 3spd I removed and the bulk of the t5 weight is positioned in about the same area as the 3spd. But, with it being longer, if there are any forces acting near the slip yoke, it will have a more dramatic effect. Can you rig a temporary crossmember in order to trouble shoot - or just go ahead and build one if you want to.
I didn't have any vibrations before or after the t5. Mine's a little different in that the yoke was only a few years old and I'm using a crossmember.
Lawrenceville, GA