Did you check out the 2-cell version of the JM35? http://www.banggood.com/Shadow-JM35-CREE-MT-G2XHP-50-1900LM-LED-Flashlight-2x26650-p-984020.html
It can be operated with protected cells (disregard the notation about unprotected cells, that refers to the one-cell version). The one-cell version is 13cm, so the 2-cell version is probably very close to your maximum 20cm criteria (I’m unable to verify actual length). And its list price of $44.99 is within your budget. (Cheaper: here)
Note: I debated the merits of this flashlight and the Nitecore EC4S long and hard before making my purchase decision. The JM35 MT-G2 2-cell had better tint, slightly more comfortable handheld form factor, longer runtimes, and seemed to have slightly better throw. The EC4S had a lighter and more compact form factor, better UI, and more “bells & whistles” (e.g., battery indicator) - albeit at the cost of vampire drain. The EC4S was $20 more expensive for me, out of pocket expenses. If I had not already owned a set of 26650 cells and a 26650 charger, the cost of the two flashlights would have been virtually the same.