Extending Morse-Smale pair from submanifolds?

Question

The following proposition is extracted from Audin & Damian's Morse Theory and Floer Homology, Proposition 4.6.3:

Let $(f,X)$ be a Morse-Smale pair on $V$ (a submanifold of $W$). Then there exists a Morse-Smale pair $(\tilde f,\tilde X)$ on $W$ that extends it and satisfies:

1. $\operatorname{Crit}(f)\subseteq\operatorname{Crit}(\tilde f)$ and the index of a critical point of $f$ is the same as its index as a critical point of $\tilde f$.
2. If $a$ is a critical point of $f$, then $\partial_{\tilde X}(a)=\partial_X(a)$, where $\partial_X\colon C_k(f)\to C_{k-1}(f),a\mapsto\sum_{b\in\operatorname{Crit}_{k-1}(f)}N_X(a,b)b$ to be the boundary operator of the Morse complex.

The proof is very sketchy and I cannot complete it. Here's what I've done:

1. We can suppose that $\dim V<\dim W$. Take a tubular neighborhood of $V$ identified with an open subset of the normal bundle of $V$, and give a (fiberwise) Riemannian metric $g$ on the normal bundle of $V$ such that any $x$ in the normal bundle satisfying $g(x,x)<2$ is contained in the chosen tubular neighborhood. We set $\tilde f(x)=f(x)+\eta(g(x,x))$, where $\eta$ is a cut off function: $\eta(x)=x$ for $x\le1/2$ and $\eta(x)=0$ for $x>1$ and $\tilde X=X-\nabla_g(\eta(g(\cdot,\cdot)))$, where $\nabla_g$ is the horizontal gradient with respect to $g$. In a tubular neighborhood of $V$, $(\tilde f,\tilde X)$ is a Morse-Smale pair and satisfies conditions.
2. We can perturb $\tilde f\in C_0^\infty$ outside a tubular neighborhood of $V$ such that it becomes a global Morse function.
3. We want to perturb $\tilde X$, but I have no idea how to proceed.

The crucial obstruction is that Smale condition is quite global so any small perturbation could destroy any virtue of preceding construction.

Any help is welcome, thanks!

POSTSCRIPT:

The construction of $\tilde X$ is flawed, since $X$ isn't defined in a tubular neighborhood. I think an extension of $X$ works, but I haven't worked out the details.