Amazing idea, I hadn't even thought of soft-clipping! I've currently written a relatively slow script that does what I describe in the OP post, after that I'll give BWA or similar a try, which is probably WAY faster, thanks for the idea!!
Edit: FWIW, here's the simple script I ended up using - ran for three hours. I haven't yet tried soft-clipping alignments yet, that comes next!
Later edit: Wow, this COMPLETELY breaks down if the downloaded sequence is large - Genbank has a few records where 10 genes are sequenced together including my 1 gene, and then the MSA breaks down and you get tiny pieces aligned all over the place, and the trimming doesn't trim much. BWA is probably better for that.
(A few minutes later: Current best pipeline I can find is MAFFT/MUSCLE with gaps turned off, followed by trimAl, in a wrapper script to parallelise? curious to hear more!)