Superior Threads has quite a bit of information on needles. Their titanium-coated topstitch needle in a size 16 would probably be the best one for your Juki:
http://www.superiorthreads.com/produ...chine-needles/
I have a Voyager, which is a stretched industrial Singer (takes a needle with a rounded upper shaft), and I have been able to buy the correct needles for it on eBay. They go up at least to a size 20 for industrial and longarm machines, and no doubt even larger for some. However, size 16 may be the biggest your Juki can take.
Try adjusting tension via the video; it should help a lot. I'm not sure about a denim needle for quilting on a frame, but in your situation I would certainly give it a try once the tensions are readjusted. With tensions adjusted, your size 14 might work too.
A domestic machine faces different stresses when placed on a frame. However, you can adjust for those stresses.
One other thing I would ask is how tight is your quilt in the frame? It's a common beginner mistake to keep the quilt too taut. Rule-of-thumb is to be able to grab a finger that is poked up from below. The quilt sandwich needs to be a little loose in the frame.