I see four ways it can be done:
Caisson down from the ice. (Needs metallurgy)
Dig in the lava tube towards the lake most of the way, stopping short.
Shatter the ice. Build a 20m high airtight box with an open bottom, pump the water out, pump lots of air into it so the pressure displaces the water, enclose your workers in it, and start digging towards the lava tube. Seal the lava tube at the lower end. Join the tubes at high pressure, then release the air pressure on the caisson. Water will flow down the lava tube and shatter your seal at the lower end.
Caissons are a pain without steam pumps (you may need a horse drawn pump or teams of people pumping), but they are possible.
Also beware of caisson disease - the bends.
Gunpowder and salt.
(Edit: you said no gunpowder in a later edit. So ignore this. I'm just leaving it up for future reference)
Tunnel towards the lake as much as you dare, stopping a few meters short. Take care to get the base on the tunnel as smooth as possible, perhaps even put a layer of concrete down.
Pack the last several meters of the tunnel with gunpowder, run a fuse, and then pack the tunnel with about a meter of salt until airtight, with solid blocks of salt in the middle, loose salt on the sides. Then put some planks of wood behind it, secured to the walls, and then plaster over it with a simple lime cement render for extra reinforcement and airtightnesss.
Light fuse. Gunpowder explodes. The salt and wood structure stops the blast from disapaiting into the tunnel, and instead fractures the rock. Water (and rocks) flow into the chamber where the gunpowder was.
The water hits the salt and dissolves it, quickly working it's way through your salt. As the salt collapses the planks and lime reinforcement gives way too, eventually you have salt water flowing down.
After a few hours the water flowing out will have cleared out the salt, planks, and most of the smaller rocks. The smooth bottom allows these to flow with the water.
Drill and Dam
Build a 25m high metal door. Lower it down to ground level below the opening. Then dig from both sides.
When finalised. Lift the door.
If you put guide rails in, you can use it as a dam to control the flow later.
Build rails and a rack line from ground level along the lava tube, and into the new tube. Build a metal drill bit the size of the tube, mount it on rails and to the rack, attach it to some gears, and have teams of people / horses pull to make the drill bit rotate, and the bit advance.
As it advances another meter. Lay another meter of track.
Because the drill bit is the size of the tunnel, it plugs the hole nicely, especially if you make it a long cylinder behind the conical part.. When water starts to seep in, keep going for the length of the cone to guarantee you've got the hole the full width, but you still have the cylinder part for safety.
Once you've reach that part, start turning the gears the other way, making the tbm retreat back down the tunnel into the lava tube, and back to the mountain entrance. Water will be seeping past you, but it should be a half decent seal and the tbm should stay on the track and rack.
At the mountain entrance, when the track and river diverge, the tbm follows the track, and the full river flows out.