Hi there,
So lets start step by step, so we can understand how you setup things at a deeper level also we can diagnose obvious mistakes first and see if it solves the problem.
So 1st you have an force propagation problem, that the steering input is working but application of forces are not propagated properly seems.
1- Short : Every thing even rocks have suspension : Having 0 suspension is not a good scenario. I understand your wording around this is a bicycle setup should have no suspension but actually in real life, cause of the air pressure inside (ForceUp), Driver Mass and gravity (ForceDown) are in balance making tires slightly go inside and thats suspension.
Furthermore general static of the bicycle frame is also a suspension cause internal rigid particles actually dampen in relation to contact points of the main frame / force points that even true for rocks (solids).
In Unreal terms , Internally suspension is actually required cause contact no contact drives how the force application works internally. A minor discontact (common in zero suspension) would cause steering/throttle disabled. Thats how the system built, so having a bit suspension 1cm is good to make sure there is some error space during the general movement. Or if you really want 0 suspension you can simply add a spoiler force pushing vehicle down at all times which is still a hack and solver wise not the best scenario.
p.Vehicle.ShowSuspensionRaycasts 1 to visualise contact raycasts.
2- Short : We cannot expect 2 wheel by default act 4 wheels. Four wheels are naturally in balance, 2 wheels balancing entirely depends on gyro force created by wheels. Ofcourse we don’t only use that for a game we use additional forces as well but to start with make CoM (Center of Mass) a bit more lower or atleast make sure its above the tire point a bit. This creates already a natural anti roll. Sometimes solver if CoM below contactcan act weirdly but you can also try lowering on your setup even more.
Additionally there is a section ArcadeControls over there there is autoRollCenter so it pushes bike up center (Assuming you are not looking for 100% phsyics simulation)
3- This is generally skipped but actually on your mesh you can search for Gyro and enable torque on wheels.
There is an external torque also on the wheels not sure but should be related to this one aswell
For the very last addition there is something called TensorScale which is overlooked sometimes but actualyl very important.
4- I see you use rear wheel class for both, is this intended ?
Cause if same class both will steer since in SS steering enabled.

You can increase inertia of the axis X Y meaning, lean & dophin movements are 2x harder to be achieved, provides damping in some terms lets say. Correct wording is Inertial mass is 2X on these axis.
Think these should change how the behaviour is already dramatically, make your wheels contact at all times, keep it upright with a better CoM, Gyro, Tensors + Arcade forces if you want.
Let me know if you try some these and the change in the behaviour of the vehicle. Even you fix we can discuss the movement aspects in further detail.