What you need to find is a technique that, given some constraint C (the left hand on the handlebar), applies some displacement to your joints so that the final animation respects those constraints while staying as close as possible to the initial animation. This could be done with constraint-baseds motion editing techniques and in particular, Per-Frame Inverse Kinematics. The idea is:
- Take an initial motion close to what you want but not perfect. Here, it is the motion in your first video;
- Determine the constraints you need to apply. Here, most probably the hand should stay on the handlebar;
- Then compute, at key frames (or similarly at every frame), the displacement you need to add to your animation to satisfy your constraints. This can be done with Inverse Kinematics for example;
- Finally, smoothly interpolate your initial animation with the final required results by smoothly "adding" the computed joint displacements.
This is not trivial to do actually and it needs some careful thinking beforehand. I suggest your read this paper which is a good explanation of the problem with different possible solutions:
Comparing Constraint-Based Motion Editing Methods, Michael Gleicher, Graphical Models, Volume 63, Number 2, page 107--134 - mar 2001
Tools for assisting with editing human motion have become one of the
most active research areas in the field of computer animation. Not
surprisingly, the area has demonstrated some stunning successes in
both research and practice. This paper explores the range of
constraint-based techniques used to alter motions while preserving
specific spatial features. We examine a variety of methods, defining a
taxonomy of these methods that is categorized by the mechanism
employed to enforce temporal constraints.We pay particular attention
to a less explored category of techniques that we term per-frame
inverse kinematics plus filtering, and we show how these methods may
provide an easier to implement while retaining the benefits of other
Your problem being in the "simple category", IK based techniques could work.
Finally, note that existing middleware could help. In particular, HumanIK from Autodesk is great for that (and much more).