The case of point wise absolute convergence for connected compact Lie groups and sufficiently smooth functions requires no deep analysis, I thought I'd write it out here. In general, the problem is basically Fourier analysis on a $\mathrm{Rank}(G)$ dimensional torus together with $|\Phi_{+}|$ summations by parts, where Rank is the dimension of the maximal torus and $\Phi_{+}$ is the positive roots.

Let $G$ be a connected compact Lie group, $T$ a maximal torus, $M$ the character lattice of the torus, $W$ the Weyl group and $\Phi$ the root system. Let $\Phi_{+}$ be the positive root system, let $M_+$ be the chamber of dominant weights and let $\rho = \frac{1}{2} \sum_{\alpha\in \Phi_{+}} \alpha$. For $\lambda \in M_{+}$, let $V_{\lambda}$ be the corresponding representation and let $d_{\lambda}$ be $\dim V_{\lambda}$. Let $\pi_{\lambda}: G \to V_{\lambda}$ be the representation map and let $\chi_{\lambda}(g) = \mathrm{Tr}\ \pi_{\lambda}(g)$ be the character. We fix a positive definite Hermitian inner product on each $V_{\lambda}$ so that $\pi(g)$ is unitary. We normalize Haar measure so that $\int_G 1 = 1$.

Let $f$ be a continuous function on $G$. Clearly, it is enough to prove pointwise convergence at the identity. Define
$$\bar{f}(x) = \int_{g \in G} f(gxg^{-1}),$$
so $\bar{f}$ is a class function.

**Lemma** Pointwise convergence of $f$ at $1$ is equivalent to pointwise convergence of $\bar{f}$ at $1$ (with the same ordering of irreps).

**Proof** Choose $v_1$, $v_2$, ..., $v_{d_{\lambda}}$ an orthonormal basis of $V_{\lambda}$. The Fourier series of $f$ at $x \in G$ is, by definition,
$$\sum_{\lambda \in M_{+}} d_{\lambda} \sum_{i=1}^{d_{\lambda}} \sum_{j=1}^{d_{\lambda}} \langle v_i,\ \pi_{\lambda}(x) v_j \rangle \int_{g \in G} f(g) \langle v_j,\ \pi_{\lambda}(g) v_i \rangle .$$
(Some $g$ and $g^{-1}$'s may be switched here.) Taking $x$ to be the identity, we have $\langle v_i, v_j \rangle = \delta_{ij}$ so the sum simplifies to
$$\sum_{\lambda \in M_{+}} d_{\lambda} \int_{g \in G} f(g) \sum_{i=1}^{d_{\lambda}} \langle v_i,\ \pi_{\lambda}(g) v_i \rangle = \sum_{\lambda \in M_{+}} d_{\lambda} \int_{g \in G} f(g) \chi_{\lambda}(g).$$
Since $\chi_{\lambda}$ is a class function, the integrals are unchanged by replacing $f$ with the class function $\bar{f}$. $\square$.

From now on, we assume that $f$ is a class function. So $f$ is determined by its restriction to $T$, which is $W$ invariant. Until I say otherwise, assume that $f$ **is sufficiently smooth that the Fourier coefficients $\int_{\theta \in T} e^{\mu}(\theta) f(\theta)$ decay faster $|\mu|^{-N}$, where $N$ is some integer depending on $G$ which you could extract from the proof**. Noting that $d_{\lambda} = O(|\lambda|^K)$ for some other constant $K$, this will make all sums absolutely convergent, so we can rearrange at will.

For $\alpha \in M$, let $e^{\alpha}$ be corresponding character of the torus. We will also sometimes abuse notation by writing things like $e_{\alpha/2}$, which is only defined on the double cover of the torus; our total expressions will be well defined. Let $\Delta :T \to \mathbb{C}$ be the Weyl denominator:
$$\Delta(\theta) = \prod_{\alpha \in \Phi_{+}} \left( e^{\alpha/2}(\theta) - e^{- \alpha/2}(\theta) \right).$$

Using the Weyl integration formula to convert integrals of class functions $G$ to integrals over $T$ and the Weyl character formula to express $\chi_{\lambda}$;, we want to show
$$f(1) = \mbox{constant} \cdot \sum_{\lambda \in M_{+}} d_{\lambda} \int_{\theta \in T} \Delta(\theta)^2 \frac{\sum_{w \in W} (-1)^w e^{w(\lambda+\rho)}(\theta)}{\Delta(\theta)} f(\theta)$$
$$= \mbox{constant} \cdot \sum_{\lambda \in M_{+}} d_{\lambda} \int_{\theta \in T} \Delta(\theta) \sum_{w \in W} (-1)^w e^{ w(\lambda+\rho)}(\theta) f(\theta) $$
$$= \mbox{constant} \cdot \sum_{\lambda \in M_{+}} \sum_{w \in W} (-1)^w d_{\lambda} \int_T e^{w(\lambda+\rho)}(\theta) \Delta(\theta) f(\theta). $$
(Here the constant absorbs $2 \pi$'s, signs and $|W|$ in some combination that isn't worth keeping track of.)

For $\mu \in M$, define $d(\mu) = \prod_{\alpha \in \Phi_{+}} \langle \mu, \alpha \rangle$. If $\mu = \lambda + \rho$, then $d(\mu) = d_{\lambda}$ by the Weyl dimension formula. If $\mu = w(\lambda+\rho)$ for $\lambda$ dominant, then $d(\mu) = (-1)^w d_{\lambda}$ by noting that $d$ is clearly $W$-antisymmetric. If $\mu$ is not of the form $w(\lambda+\rho)$ for any dominant $\lambda$ and any $w$, then $\mu$ is on one of the hyperplanes $\langle \ , \alpha \rangle =0$, so $d(\mu)=0$. In short, we can make the change of variable $\mu=w(\lambda+\rho)$ and write
$$\mbox{constant} \cdot \sum_{\mu \in M+\rho} d(\mu) \int_T \Delta(\theta) f(\theta) e^{\mu}(\theta).$$

I pause to record what we are doing for $SU(2)$. For $f$ an even, $2 \pi$ periodic function, we are trying to show that
$$f(0) = \frac{-1}{4 \pi} \sum_{n=-\infty}^{\infty} n \int_{\theta=0}^{2 \pi} f(\theta) (e^{i \theta} - e^{- i \theta}) \cdot e^{i n \theta} d \theta. \quad (\ast)$$
Assuming that the Fourier cofficients are small enough to rearrange this sum, we can regroup as
$$\frac{-1}{4 \pi} \sum_{m=-\infty}^{\infty} \int_{\theta=0}^{2 \pi} f(\theta) ((m-1) - (m+1)) e^{i m \theta} \quad (\ast \ast)$$
which is clearly the standard Fourier tranform. Note, though, that the rate of decay of Fourier coefficients which justifies the rearrangement is more stringent than that which makes the Fourier sum absolutely convergent.

We claim that the same thing happens in general. Expanding the product for $\Delta$, we can rewrite the sum as

$$\sum_{\nu \in M} \int_T f(\theta) \sum_{\epsilon: \Phi_{+} \to \{ \pm 1 \}} \prod_{\alpha \in \Phi_{+}} \epsilon(\alpha) \cdot d\left( \nu - \sum_{\alpha \in \Phi_{+}} \epsilon(\alpha) \alpha \right) e^{\nu}(\theta)$$

Now, $d(x)$ is a polynomial of degree $|\Phi^{+}|$, and we have applied $|\Phi|$ partial difference operators to it. So $\sum_{\epsilon: \Phi_{+} \to \{ \pm 1 \}} \prod \epsilon(\alpha) d(x- (1/2) \sum \epsilon(\alpha) \alpha)$ is a constant, and I leave it as a challenge for the reader to compute that it is $|W|$. So our sum collapses to
$$\mbox{constant} \cdot \sum_{\nu \in M} f(\theta) e^{\nu}(\theta).$$
Because the world is just, the constant works out.

Stated more conceptually, $\Delta(\theta) e^{\nu}(\theta) f(\theta)$ is a $|\Phi_{+}|$-fold partial difference of $\int_{\theta} e^{\nu}(\theta) f(\theta)$, and $d(\nu)$ is a degree $|\Phi_{+}|$ polynomial. If we can justify summing by parts enough times, we should be able to turn
$$\sum_{\nu} \int_{\theta} \Delta(\theta) e^{\nu}(\theta) f(\theta) \cdot d(\nu)$$
into
$$\sum_{\nu} \int_{\theta} e^{\nu}(\theta) f(\theta) \cdot 1.$$

I'm pretty sure that I learned somewhere that, if $\sum A_i (B_{i+1} - B_i)$ is $(C,k)$ Cesaro summable, then the summation by parts $\sum (A_{i-1} - A_i) B_i$ is $(C,k+1)$ Ceasro summable. Taking $A_n=n$ and $B_n = \int_{\theta=0}^{2 \pi} f(\theta) e^{i n \theta} d \theta$, we see that $(\ast)$ should be $(C,2)$ Cesaro summable.

Unfortunately, on higher dimensional torii, the question of what the analogue of Cesaro summation is a mess.