Is that a nylon washer stuck to the end of your tube? Or is is a nylon bushing in the tube?
They make two types of syringes, Luer-lock and flusshing. The former is designed to accept a needle, the latter is designed to irrigate tissues. Try to find a small version of the latter.
I have also found that the grease should be a light weight grease, NOT a bearing grease. You should not have to warm the grease to fill the syringe, nor to inject it into the tube. If you use a bearing, or heavy grease and have to warm it to inject it, it will harden in the shaft and create extra drag on your shaft. remeber that the water is going to cool the tube, the grease and the shaft. DRAG. A light weight teflon bearing grease will seal the shaft and lubricate the the shaft in the tube with minimum drag.
My last point is that if you use a heavy grease, it will be so much harder to remove it when the time comes.😀