Got the same problem for both BF3 and 4.
I can uninstall "battlelog web plugins" from "Programs" and install the battlelog-web-plugins_2.3.0_119.exe ... BF still insist that I have to install it.
I have checked that the 2 plugins are running in the browser.
I had the same problem with IE10 and the only way to play BF was to use Firefox for me so I recommend that you try another browser if the one that you are currently trying to use is not playing well with the Battlelog plugin.
Battlefield Player on PC, Xbox One, PS4, Xbox 360, PS3
BF3, BF4, BF1 DICE Friend tag awardee
SWTOR Player on the Harbinger and Red Eclipse