Efficient algorithm for edit distance for short sequences
One approach is to build a Levenshtein automaton for the fixed string (see, e.g., here).  Given a string $x$ and a distance $D$, you can build a DFA that recognizes all strings that are at distance $\le D$ from $x$.  Thus, you can test whether a string is close to $x$ in $O(n)$ time, where $n$ is the length of the string.  I'm not sure what the space requirements are to store the DFA (they are linear in $m,n$, but might be exponential in $D$).

Alternatively, you could use an "early-out" algorithm for computing the edit distance.  You mentioned that you are only interested in the edit distance if it is less than some threshold $D$.  There is an "early-out" algorithm for computing the edit distance whose running time is $O(\max(n,m) \times D)$, which computes the edit distance if it is $\le D$ or else outputs "too big" if it is $&gt;D$.  Basically, you do the standard dynamic programming algorithm for the edit distance, but only compute the elements of the matrix that are $\le D$ away from the diagonal.  In your case this might or might not be better than the other alternatives.