Any gain of such a construction would have been minimal. If you look at the graphic from the Wikipedia article on the Road of Life, it shows relative positions held.
The Finns already held the entire Northern boundry of the lake, so would make no gain in mobility by such a construction. The Germans actually held territory much closer to the ice road, so since the nearest Finnish controlled areas appear to be three to four times farther away, again such a construction would not give them any appreciable gain on the ability to assault the Soviet road.
There is also an unsupported entry on the wiki page:
The Finnish forces intentionally left the supply route open in tacit
defiance of Germany's requests.
This implies a (possible) resistance on the part of the Finns to fully participate in the war beyond what they deemed in their own interests (regaining losses from the Winter War).
So bottom line is that there would be little or nothing militarily to gain by such a construction.