at.algebraic topology - primitive of an exact differential form with special properties

I think, the answer can depend on how you interpret the question. Let me show that the answer is negative for one of the interpretations already in the case of $2$-dimensional manifolds. We study the question locally in a neighbourhood of a vertex of a triangulation, so the condition on the integral over $n$-simplexes does not play any role. The obstruction for the existence of $beta$ comes form the local behaviour of curves at a vertex.

Lemma. Let $alpha=dxwedge dy$ on $mathbb R^2$. For $nge 8$ there exist $gamma_1,...,gamma_n$, smooth rays on $mathbb R^2$ that meet at $0$ with different tangent vectors and such that there is no $beta$ defined in any neighbourhood of 0 with $dbeta=alpha$ and vanishing been restricted to $gamma_i$.

It is clear that this lemma implies the negative answer to a version of the question, when we are not allowed to deform the triangulation.

Proof of Lemma. Suppose by contradiction that $beta$ exists. Then $beta_1=beta-frac{1}{2}(xdy-ydx)$ is a closed 1-form. So we can write $beta_1=dF$, where $F$ is a function defined in a neighbourhood of $0$, $F(0)=0$. Since the number $n$ of the rays is more than $2$, $dF$ should vanish at zero. Moreover, it is not hard to see, that since the number of rays is more than $4$, the quadratic term of $F$ vanishes at zero too.

Now, since $beta$ vanishes on $gamma_i$, the restiction of $beta_1$ to $gamma_i$ equals $frac{1}{2}(ydx-xdy)$. So we get the formula for $F$, resticted to $gamma_i$
$$F=frac{1}{2}int_{gamma_i}ydx-xdy.$$
Now, we will chose the rays $gamma_1,...,gamma_8$. Namely $gamma_1(t)=(t,t^2)$, $gamma_2(t)=(t,t-t^2)$, and take $gamma_3,...,gamma_8$ by consecutively rotating $gamma_1,gamma_2$ by $pi/2$, $pi$, $3pi/2$.

It is not hard to see, that $F$ is cubic modulo higher terms in $t$ when it is restricted to $gamma_i$. At the same time $F$ is positive on $gamma_{1},gamma_3,gamma_5, gamma_7$ and negative restricted to other rays. So it changes its sign at lest $8$ times on a little circle surrounding $0$. This is impossible for a cubic Function (in a little neighbourhood the cubic term of $F$ should be dominating). Contradiction.