Let the maximum length sequence $$s_n=(-1)^{Tr(\beta \alpha^n)}$$have period $2^m-1,$ and be nontrivial with $\beta\neq0.$ Your sum is $$\sum_{n=1}^N s_n \exp(2 \pi i n/N).$$

If $\beta=0,$ you have the standard unmodulated linear exponential sum with the bound $$\min\left\{N,\frac{1}{|\sin \pi/N|}\right\}. $$

If $N\leq m,$ all possible $N-$tuples in $\{\pm 1\}^N$ are taken on by $$(s_n,\ldots,s_{n+N-1})$$ so a better bound is not possible.

In the intermediate range $m<N\leq 2^m-1,$ maybe a bound along the lines suggested by @FelipeVoloch is possible, I am not sure.

For example direct use of van der Korput inequality looks difficult since the partial correlations themselves are sums of the form we are trying to bound.

Let the maximum length sequence $$s_n=(-1)^{Tr(\beta \alpha^n)}$$have period $2^m-1,$ and be nontrivial with $\beta\neq0.$ Your sum is $$\Gamma(m,N)=\sum_{n=1}^N s_n \exp(2 \pi i n/N).$$

If $\beta=0,$ you have the standard unmodulated linear exponential sum with the bound $$\min\left\{N,\frac{1}{|\sin \pi/N|}\right\}. $$

If $N\leq m,$ all possible $N-$tuples in $\{\pm 1\}^N$ are taken on by $$(s_n,\ldots,s_{n+N-1})$$ so a better bound is not possible.

Edit: In the intermediate range $m<N\leq 2^m-1,$ a bound along the lines suggested by @FelipeVoloch is indeed possible.

From Theorem 8.78 in the book Lidl and Niederreiter, Finite Fields (Encyclopedia of Mathematics and its Applications, we have $$ |\Gamma(m,N)|\leq \sqrt{2^m}, $$ which is nontrivial if $N>\sqrt{2^m}$.

Let the maximum length sequence $$s_n=(-1)^{Tr(\beta \alpha^n)}$$have period $2^m-1,$ and be nontrivial with $\beta\neq0.$ Your sum is $$\sum_{n=1}^N s_n \exp(2 \pi i n/N).$$

If $\beta=0,$ you have the standard unmodulated linear exponential sum with the bound $$\min\left\{N,\frac{1}{|\sin \pi/N|}\right\}. $$

If $N\leq m,$ all possible $N-$tuples in $\{\pm 1\}^N$ are taken on by $$(s_n,\ldots,s_{n+N-1})$$ so a better bound is not possible.

In the intermediate range $m<N\leq 2^m-1,$ maybe a bound along the lines suggested by @FelipeVoloch is possible, I am not sure.

Let the maximum length sequence $$s_n=(-1)^{Tr(\beta \alpha^n)}$$have period $2^m-1,$ and be nontrivial with $\beta\neq0.$ Your sum is $$\sum_{n=1}^N s_n \exp(2 \pi i n/N).$$

If $\beta=0,$ you have the standard unmodulated linear exponential sum with the bound $$\min\left\{N,\frac{1}{|\sin \pi/N|}\right\}. $$

If $N\leq m,$ all possible $N-$tuples in $\{\pm 1\}^N$ are taken on by $$(s_n,\ldots,s_{n+N-1})$$ so a better bound is not possible.

In the intermediate range $m<N\leq 2^m-1,$ maybe a bound along the lines suggested by @FelipeVoloch is possible, I am not sure.

For example direct use of van der Korput inequality looks difficult since the partial correlations themselves are sums of the form we are trying to bound.