What's interesting is that some folks have been cleaning them in a similar fashion for a while but didn't think to fill that way... and I never thought to clean that way until someone pointed out the obvious. Interesting, how the mind works.
LL syringes are available from all kinds of vendors who sell nicquid. Start by checking the pages of one of your current vendors and just drop a couple in the cart on your next order. For filling, 3ml is just about ideal (fits inside of most 30ml bottles). For
cleaning you'll want a bigger one, at least 10ml though 20 to 30ml will reduces cycles. Up to 60ml are pretty readily available with a little Googling. If you can shop by brand BD and Monoject are the two most commonly found; I recommend BD because their lock collars tend to be smooth rather than knurled.
Local sources might include a pharmacy, hobby shop or hydroponics shop (which sometimes carry up to 140ml but that's just crazy big). Knowing a doctor, nurse or veterinarian might also be an in you can use.