The answer to your question is yes.

The subgroups of ${\rm SL}(2,q)$ were classified by Dickson in 1901 (and probably earlier). For more general questions about subgroups of ${\rm GL}(n,q)$, there is a theorem of Aschbacher that classifies them into nine types, and is used as the basis of algorithms for identifying these subgroups.

Using these results, we find that a proper subgroup of ${\rm SL}(2,q)$ with $q=2^n$ must be either reducible, imprimitive and isomorphic to $D_{2(q-1)}$, semilinear and isomorphic to $D_{2(q+1)}$, or conjugate to a subgroup of ${\rm SL}(2,r)$ where $r = 2^m$ with $m|n$.

For testing reducibility there is the efficient $\mathtt{MeatAxe}$ algorithm.

There are general tests for imprimitivity and semilinearity, but in this case it would probably be easiest to find look for an abelian subgroup of index $2$.

As YCor mentioned in the comments, you can test for definability over a subfield. Usually you start by calcuating the characteristic polynomials of lots of random group elements, and if their coefficients all lie in a subfield then it is is very likely that the whole group does. There is a method of finding the conjugating matrix to verify this, but I would have to look that up.

There are two software packages with extensive facilities for these types of calculations, GAP and Magma. I am more familiar with Magma, so I will mention some of the commands that you could use to do this. $\mathtt{IsIrreducible}$, $\mathtt{IsPrimitive}$, $\mathtt{IsOverSmallerField}$, and and $\mathtt{IsSemiLinear}$ do what you might expect.

It is possible that if $q$ is very arge then you could run into some problems related to discrete log calculation, but I don't think so if you just want a yes or no answer.

The answer to your question is yes.

The subgroups of ${\rm SL}(2,q)$ were classified by Dickson in 1901 (and probably earlier). For more general questions about subgroups of ${\rm GL}(n,q)$, there is a theorem of Aschbacher that classifies them into nine types, and is used as the basis of algorithms for identifying these subgroups.

Using these results, we find that a proper subgroup of ${\rm SL}(2,q)$ with $q=2^n$ must be either reducible, imprimitive and isomorphic to $D_{2(q-1)}$, semilinear and isomorphic to $D_{2(q+1)}$, or conjugate to a subgroup of ${\rm SL}(2,r)$ where $r = 2^m$ with $m|n$.

For testing reducibility there is the efficient $\mathtt{MeatAxe}$ algorithm.

There are general tests for imprimitivity and semilinearity, but in this case it would probably be easiest to find look for an abelian subgroup of index $2$.

As YCor mentioned in the comments, you can test for definability over a subfield. Usually you start by calcuating the characteristic polynomials of lots of random group elements, and if their coefficients all lie in a subfield then it is is very likely that the whole group does. There is a method of finding the conjugating matrix to verify this, but I would have to look that up.

There are two software packages with extensive facilities for these types of calculations, GAP and Magma. I am more familiar with Magma, so I will mention some of the commands that you could use to do this. $\mathtt{IsIrreducible}$, $\mathtt{IsPrimitive}$, $\mathtt{IsOverSmallerField}$, and and $\mathtt{IsSemiLinear}$ do what you might expect.

It is possible that if $q$ is very large then you could run into some problems related to discrete log calculation, but I don't think so if you just want a yes or no answer.