A whole tube of grease? Wow...I would recommend pulling your wheel/hub off and pray that you didn't push either the seal out the back or push grease past the seal into the brake area.
I worked at servicing the munitions trailers used by the Air Force for a very large part of my career. You only need to pull the bearings, clean them, inspect them, repack with (since Uncle Sam isn't paying) High Quality bearing grease, and re-install. If you want, slather a little extra grease over the bearing so it kind of has a glazed donut look. But that is all you really need. The Air Force requires a 720 day inspection of the bearings on the trailers. The ones that went bad, were the ones that sat and were hardly used.
While it is your equipment and trailer, and you can do as you will, I would personally advice against using the "bearing buddy" type greasing systems unless it is on a boat trailer, for the reasons stated in the posts above. It is very easy to over-grease your wheels and end up with a bad situation inside the hub.
Just my two cents...Your mileage may vary...Stepping off my soapbox.