I believe your confusion may stem from your problem statement. You mention a point mass in uniform circular motion and deriving the equation.
First, are you trying to derive the equation (i.e. expression) for the centripetal force, or the centripetal acceleration. If you are attempting to derive the expression for the centripetal acceleration, you do not need (and cannot use) force, or energy as useful concepts. You only need the definitions of position, velocity, and acceleration as vectors and motion in the plane. The simplest route (if you know calculus) is to describe the circular motion (position) as
x = R cos (omega * time) I + Y sin (omega * time) J, where I and J are unit vectors in the x and y directions. Then take two derivatives to get
a = d2x/ dt2 = (-) R * omega * omega (cos (omega * time) I + Y sin (omega * time) J) = - R omega * omega times a unit vector in the radial direction cos (omega * time) I + Y sin (omega * time) J.
So acceleration is radially inward (note the minus sign) and has magnitude R * omega squared. Hence the name centripetal which means "center seeking"
because it is directed towards the center.
Without calculus you can still derive the centripetal acceleration kinematically, but you need to remember similar triangles from high school geometry, and be clever.
The centripetal force is merely the point mass (mass) times the centripetal acceleration. Energy plays no role at all.
I have found (and I use to tell my students), that the best idea is to recall Newton's laws.
1. A body at rest tends to remain at rest and a body in motion tends to remain in motion in a straight line unless acted upon by a force.
Your proposed point mass is in circular motion so it has a force acting on it.
What provided the force?
You did not say. In could be the Earth's gravity if you are treating a satellite. It could be electrical if you are treating an electron in orbit around a proton in a hydrogen atom and using the old quantum theory, it could be your muscles if you are spinning the rock on a string. Etc.
2. It turns out you only have half a problem here without defining the force holding the point mass in circular motion.
By the way force is a vector and momentum is a vector.
Newton's second law (often misquoted as F = ma) is actually F = dp/dt. This law holds for all three components x, y, and z separately so
Fx = d px / dt; Fy = d py / dt. and Fz = d pz / dt.
Your construct "intensity of momentum" , which I take you mean the "magnitude" of the momentum vector is a concept that Newton did not address, and I cannot see where it is useful. Surely, Newton's second law holding for each component separately is a tighter requirement and will be more useful.
3. Insofar as energy is concerned, energy is a scalar, "centripetal" meaning center seeking implies a direction. Centripetal force and centripetal acceleration makes sense, but centripetal energy does not.
4. I sort of blame the misconception that energy is needed to "preserve" motion on Star Trek.
The starship Enterprise was always in danger of plummeting to the planet when their dilithium crystals gave out. In reality, a spacecraft in orbit without a source of energy could take years to decay from even a modest 100 mile altitude. (The decay is eventually from frictional losses of the spacecraft energy from the few atmosphere molecules) The Earth has been in orbit around the Sun for billions of years, (and without dilithium crystals) too.
5. Actually though, your thoughts on energy transfer did motivate me to revisit some concepts on energy transfer, that I remember reading more than 30 years ago in an unexpected source. I encourage any reader to examine what is written in "The Classical Electromagnetic Field" by Leonard Eyges. Dover edition 1972. The interested reader need not have the prerequisites in EM theory to explore the last paragraph on page 201,and first two paragraphs on page 202, referring to energy transfer. I felt it is very revealing.
I hope this lengthy reply helps and especially provides some humor.
