= Solution
The displayed torus-only <Weyl integration formula> needs $\varphi$ to be a <class function>, meaning $\varphi(xgx^{-1})=\varphi(g)$. This hypothesis is implicit in the usual formula but is not printed here. We prove the general version for a continuous function, then specialize:
$$
\int_G\varphi(g)\,dg
=\frac1{|W|}\int_T\int_{G/T}\varphi(xtx^{-1})\,d\mu(xT)\,|\Delta(t)|^2\,dt.
$$
Here $dg$ and $dt$ are probability <Haar measures>; $d\mu$ is the invariant probability measure on the <homogeneous space> $G/T$; and $W=N_G(T)/T$ is the finite <Weyl group>.
We use the following standard maximal-torus facts, as permitted: every element of a connected <compact Lie group> lies in some <maximal torus>; all maximal tori are conjugate; two elements of a fixed $T$ are conjugate in $G$ exactly when they are in the same $W$-orbit; and the complexified <Lie algebra> decomposes into $\mathfrak t_{\mathbb C}$ and one-dimensional <root spaces>. Opposite <roots of a root system> give real two-dimensional planes in $\mathfrak t^\perp$. At an element $t$ for which no root character has value one, the identity component of its centralizer is $T$.
Choose a positive <root system> $\Phi^+$. Regard each root $\alpha$ as a <character> $\alpha:T\to U(1)$ through the <Adjoint representation>. A convenient globally defined denominator is
$$
\Delta_0(t)=\prod_{\alpha\in\Phi^+}(1-\alpha(t)^{-1}).
$$
The usual <Weyl denominator> additionally has a unit-modulus factor $e^\rho(t)$, with $\rho$ half the sum of positive roots. Where that half-weight is not a global torus character, it may be interpreted on a covering torus; the squared modulus is nevertheless unambiguous. In either convention,
$$
|\Delta(t)|^2=|\Delta_0(t)|^2
=\prod_{\alpha\in\Phi^+}|1-\alpha(t)|^2.
$$
Average an <inner product> on $\mathfrak g$ over the <Adjoint representation> to make it invariant. It defines a bi-invariant <Riemannian metric> on $G$ and the quotient metric on $G/T$. Consider the conjugation map
$$
q:G/T\times T\to G,\qquad q(xT,t)=xtx^{-1}.
$$
At $(T,t)$, identify the source tangent directions with $X\in\mathfrak t^\perp$ and $Y\in\mathfrak t$, and translate the target tangent vector to the identity by $t^{-1}$. Differentiating gives
$$
dq_{(T,t)}(X,Y)=(\operatorname{Ad}(t^{-1})-I)X+Y.
$$
The two summands are orthogonal. On a real root plane, $\operatorname{Ad}(t^{-1})$ is a rotation with complex eigenvalues $\alpha(t)^{-1},\alpha(t)$. Thus the absolute real <Jacobian determinant> on this plane is $|1-\alpha(t)|^2$. The torus direction contributes one. Therefore the <conjugation Jacobian for a compact Lie group> is
$$
\boxed{J_q(t)=|\Delta(t)|^2.}
$$
This is also the Jacobian for the normalized measures: the Riemannian submersion has fibers isometric to $T$, so $\operatorname{vol}(G)=\operatorname{vol}(G/T)\operatorname{vol}(T)$; the probability-volume normalization factors cancel.
Let $T_{\rm reg}$ be the elements for which $J_q\neq0$. The restriction of $q$ is a local <diffeomorphism> onto the regular elements of $G$. It is $|W|$-to-one. Indeed, fixing $t_0\in T_{\rm reg}$, any pair with $xtx^{-1}=t_0$ has $xTx^{-1}=T$, because conjugation identifies the identity components of the two centralizers. Thus $x\in N_G(T)$, and there is exactly one pair $(xT,x^{-1}t_0x)$ for each coset in $N_G(T)/T$. This argument does not require the full centralizer of $t_0$ to be connected.
The nonregular locus has measure zero. In $T$ it is a finite union of sets $\alpha(t)=1$ of smaller dimension; their conjugation images are images of smaller-dimensional manifolds under $q$, and hence have zero volume in $G$. The change-of-variables formula for the regular covering therefore gives
$$
|W|\int_G\varphi(g)\,dg
=\int_T\int_{G/T}\varphi(xtx^{-1})\,d\mu(xT)\,|\Delta(t)|^2\,dt.
$$
For a <class function>, the inner integral is $\varphi(t)$, proving the required formula with its correct hypothesis.
Without conjugation invariance, the torus-only expression can be false. For $G=SU(2)$ with diagonal $T$, take $\varphi(g)=|g_{12}|^2$. It vanishes on $T$, whereas its group integral is $1/2$: the two squared coordinates of a Haar-distributed unit column have equal means and sum to one. Thus the conjugacy average cannot simply be dropped.
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