I've never seen a FD cage that requires a 4-5mm gap between the cage and big ring. Usually it's 1-3mm.

There is a way to increase the cable tension, without using an in-line adjuster. I owned two LOOK 585 bikes that have no frame mounted cable adjusters and never resorted to installing an in-line unit. Shift the bike to the largest cog and little ring. First, properly adjust the limit screw to eliminate any chain rub and produce no more than 1mm of additional cage to chain clearance, without the cable attached. After that, deliberately turn the low limit screw CW about 1/2 turn. Use pliers to grab the cable, pull it tight and tighten the cable clamp bolt. Then turn the limit screw CCW, about 1/2 turn, so it's back to it's original setting.

What this does is swing the end of the lever arm down, while you attach the cable. This will increase the cable tension, when the limit screw is returned to it's normal position. The exact amount to turn the screw take a bit of trial and error. The amount will be greater with a brand new cable, than it would be at some later point, when the cable tension needs a slight tweak.