One of the American fountain pen men of repute, Richard Binder I think said of the fountain pen generally that it is nothing more than a controlled leak.
The basis of the control is that the ink is drawn by capillary action onto the paper from the reservoir, via the nib and feed and as that ink is used, air is admitted into the reservoir in an equally controlled manner to maintain ink flow without causing the ink in the nib to either starve or flood. The reservoir maintains a pressure just very slightly less than atmospheric, so resisting leaks when the pen is not used.
I have restored several old pens, a lot of them 50s and 60s piston fillers. Many of these show exactly the symptoms yours show because air leaks past the piston seals either in storage or in use and, with air getting into the reservoir through another means, the pen doesn't fill well and the feed floods and ultimately the nib drips ink too.
I expect you may have a tiny air leak in the piston assembly, this explains both the partial fill and unexpected emptying of the pen! You'll probably fix it with a little lubrication.
The best stuff is silicone grease as you mention, supplied with TWSBI pens or available in little tubs for lubricating rubber diving gear. Silicone won't mix with ink or rot the seals so much better than vaseline or lithium grease. I tend to apply a tiny amount, half a grain of rice size with a tooth pick on pens like yours smeared round the inside of the barrel just below the piston seal and then work the piston up and down.
It could be the seals have a flaw which lubrication won't fix but I have found a dry seal can be bought back to life with lubrication, saving disassembly and replacing parts.
The pen looks like a replica of the TWSBI Diamond 540/ 580. A great pen in it's own right.