Get a numerical bounce solution and avoid boundary values problems?

Questions
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1

I want to solve numerically the following differential equation

y''(x) + \frac{3}{x}y'(x) = \frac{d U(y)}{y}\,,\qquad U(y) = \frac{1}{4}y(x)^2(y(x)-1)^2 -\frac{1}{400}y(x)^3

in the region x\in[0,+\infty] with the bounce boundary conditions

y'(0) = 0\,,\qquad y(+\infty) = 0.

I expect a flat direction around x=0, so substituting y'(0)=0 and y''(0) = 0 in the differential equation, I get an estimate for the initial condition y(0) \sim 1.

We can try to guess the form of the solution by making some approximations. Indeed,
I can solve analytically the differential equation for big values of x\geq R \gg 1, by neglecting the 1/400y(x)^3 term in the expression of U(y), as y(0) \sim 1 and y(+\infty) = 0; then, by defining U_0(y) = \frac{1}{4}y(x)^2(y(x)-1)^2 we can solve

y''(x)= \frac{d U_0(y)}{y}\,

from which we get

y_R(x) = \frac{1}{2}\left(1-\text{tanh}\left[\frac{x-R}{2\sqrt{2}}\right]\right).

So, roughly the approximated solution is

y(x) = \left\{\begin{array}{c c} 1 & x \ll R\\ y_R(x) & x \geq R \end{array}\right.

and it is shown in the following picture for R=400

enter image description here

So, I have tried to solve numerically this Cauchy problem without any approximation by setting a cut-off R on the x coordinates and using the following code

V[y_[x_]] = 1/4 (-1 + y[x])^2 y[x]^2 - 0.0025 y[x]^3

Psol = ParametricNDSolveValue[{D[y[x], {x, 2}] + 3/x D[y[x], x] == D[V[y[x]], y[x]], y'[10^-5] == 0, y[10^-5] == tp, WhenEvent[y[x] < 0 || y[x] > tp, "StopIntegration"]}, y, {x, 10^-5, R}, {tp, R}]

ListPlot[Table[{tp, Abs[Psol[tp, 50][50]]}, {tp, 0.8, 1.015, 0.001}]]

where the last line may be useful to determine the magnitude of tp such that I recover a vanishing asymptotic value. However, I get the following plot

enter image description here

So, I guess the method is not suitable for this problem. In particular, the solution Mathematica finds does not respect the asymptotic boundary condition y(+\infty) = 0, and moreover it starts oscillating after some time like in the following picture and does not go to zero at infinity

enter image description here

How can I get the numerical solution?