After the Harbor Freight saw failed (surprise!), I experimented a bit. Normal saw blades dull very quickly. I tried carbide grit blades in the reciprocating saw which worked very well. The blades were too deep to cut the corners required in spots so I modified a blade as shown. These blades last a long time (pack of 3 are probably all I'll need for the whole project) and work really well.
A good shop vac with a bag filter for fine particulate matter is required. The normal foam filter just blows the nasty fiberglass stuff all over the place.
airbody saw and reciprocating saw
Regular carbide blade and modified version
A good shop vac and a remote location help with the mess