Maybe a spring line
Have you tried rigging a spring line from the outer most cleat on the finger dock to an onboard cleat just aft of amidships? I would think if you pick up and secure that line as soon as possible upon entering the slip you could apply forward power against it and steer to port using the spring line to "spring" and hold the bow in to port while kicking the stern (against the current) to starboard. (It would take a bit of trial and error adjusting to get the length and position of the line just right, of course.) You might then be able to attach other docking lines and warp/ease the rest of the way in. Sounds like you've got quite a challenge here.