I began by welding up the interior assembly. I watched the videos a couple of times, then duplicated the workflow using the dry components. I tried an 18G syringe to start the first join. Glue shot out all over the place. Needle too large!
Swapping down to a 22G, and snipping off the point with my multitool worked just fine. Glued up the assembly with no problems at all. (apart from blemishes, finger prints and glue runs)
The syringe would have less fine control than a bottle. Stiction in the plunger makes for an inconsistent flow. Still the results look ok to me.