Even in the more restrictive category of smooth manifolds, there is no 'canonical' way to do this, if by 'canonical' one means 'invariant under diffeomorphims'. However, in the category of smooth $n$-manifolds endowed with an affine connection, there is such a local 'splitting' defined by the connection, based on the fact that such a connection defines, for each $p\in M$ a diffeomorphism from a neighborhood of $0_p$ in $T_pM$ to a neighborhood of $p\in M$, so that one can use the 'linearity' of the vector space $T_pM$ to define the local splitting by Taylor series based at $0_p$.

If you look at the complex curves $w=z^k$ in $\mathbb{C}^2\simeq\mathbb{R}^4$, which are calibrated and therefore area-minimizing, you'll find that the area enclosed in a fixed ball about $0$ goes to infinity as $k$ goes to infinity. Doesn't this imply that there can't be an $f$ such as you describe when $(k,n)=(2,4)$?

@IanGershonTeixeira: This is not true for $\mathrm{SU}(2)$ endowed with a generic left-invariant metric. It is true for a biïnvariant metric, but that's a very special case.

@AndreasStergiou: Well, those values of $\Delta^T = 24a_0$, namely $[46.17, 46.23, 46.91]$, which I took from your Section 5.2 $(N=4)$ already don't match what can happen for solutions that can be written in 'bi-quadratic form: $$[0, 8, 16, 19.20, 24, 27.20, 32, 32.09, 32.73, 35.20, 38.40, 40, 40.09, 40.73, 41.16, 47.83, 48].$$ Interesting.

@AndreasStergiou: Thanks. I'll have a look at your results. Have you computed the values of $\Delta^2T$ for these 'uneven' solutions? I would be interested to know those.

Oops! That last comment should have ended "one pair of which are the two complex roots of the of the polynomial satisfied by the irrational value". (I'm responding to comments while I'm traveling, and maybe I'm a little distracted.)

I'll just add that the polynomial satisfied by $\Delta^2(T)$ must be of high degre as $n$ increases. All of the roots at a given value of $n$ occur for all higher values of $n$. For $n=3$, the polynomial has some roots of high multiplicity (probably corresponding to the fact that the corresponding orbit meets the codimension 3 subspace in several distinct points (which are still all rational when $\Delta^2(T)$ is rational!). Moreover, it has 2 distinct pairs of complex conjugate (non-real) roots, one pair of which are the two complex roots of the polynomial satisfied by the rational value.