Something must be incorrect in the install if the Aican didn’t fit through the frame. The Aican housing is significantly thinner than standard housing and should have no problem going through. I assume it’s the ferrules that you are not able to get to pass through. It took a little patience to get everything through the frame with all the pieces in the right place. Here’s how I did it.
First I ran a section of standard housing through the frame. I then put the brake cable into the lever but did not feed it through the housing yet. Then I put an Aican ferrule onto the brake cable, pushed it up to the lever and put that cable into the housing that was in the frame. Then I pulled the regular housing out from the back of the frame leaving the ferrule at the lever and the raw cable running through the frame. Then working from the back of the frame I started putting on links of Aican. As I filled up the cable with them they eventually came out the front of the frame and at that point I slid the ferrule from the lever down onto the housing. I continued to install pieces until the housing was the length that I needed. Then I put on the ferrule at the brake end and clamped the cable in the brake.