Sard's Theorem, which is foundational, may be an example of a theorem of the sort you are looking for when the differentiability class of the function is low. Let's recall its classical statement:
Sard's Theorem: Let $f\colon\, \mathbb{R}^n\to\mathbb{R}^m$ be a $k$ times continuously differentiable function, where $k\geq \text{max}(n-m+1,1)$. Let $X$ be the critical set of $f$. Then $f(X)$ has Lebesgue measure $0$ in $\mathbb{R}^m$.
A constructivist version was proven by Yuen-Kwok Chan in 1971:
Chan, Yuen-kwok, A constructive proof of Sard's theorem. Pacific J. Math. 36: Let $f\colon\, \mathbb{R}^n\to\mathbb{R}^m$ be a $k$ times continuously differentiable function, where $k\geq \text{max}(n-m+1,1)$. Let $X$ be the critical set of $f$. Then $f(X)$ has Lebesgue measure $0$ in $\mathbb{R}^m$.
A constructivist version was proven by Yuen-Kwok Chan in 1971:
Chan, Yuen-kwok, A constructive proof of Sard's theorem. Pacific J. Math. 36, 291–301 (1971; MR0276988).
The constructivist version relaxes the statement of Sard's theorem in a benign way ("critical points" are replaced by "almost critical points") and in at least one less benign way:
The function $f$ is taken to be a $k$ times continuously differentiable function, where $k\geq 2+\frac{1}{2}(n-m)(n-m+1)$.I can imagine this being a real issue, because the function actually given to you might be $C^k$ only for $\text{max}(n-m+1,1)\leq k< 2+\frac{1}{2}(n-m)(n-m+1)$. John Milnor on Mathematical Reviews asks whether the bound on $k$ can be tightened, and so does the author at the end of the paper. If not, then Sard's Theorem for small $k$ seems to me to be a genuinely important result which can be proved classically, but not constructively.