While tracing some errant electrical current, I ended up in the big concrete riser above the septic tank. The electrical is in there, rather than outside on a pedestal as is normal now. There, I discovered (among other, scarier things) that the cable and leads to the septic alarm are connected to nothing. They're just wire nutted and dangling.
EDIT: That power lead appears to be properly connected to the tank alert inside the house. (Need to confirm this.)
Inside the house the alarm unit is an SJE Tankalert XT. (https://www.sjerhombus.com/products/tank-alert-xt/) It normally takes this type of float switch (a normally open one) (https://www.sjerhombus.com/products/sje-signalmaster/)
None of these were available, so I purchased the closest thing I could find, a barracuda (branded also SJE Rhombus Universal float switch), and I attached an outlet to the leads, confirmed I had power, then plugged this float in, then manually tipped it up and down. I could feel the ball bearing switch inside moving to the closed and open positions, but the alarm inside never sounded.
Hitting "test" on the box generates the audible alarm (but no flashing light), for whatever that's worth.
Since it's nearly impossible to get a plumber or septic repair person out, I need to get this done myself, but I'm a bit lost on whether the float switch wasn't suitable, or whether the tankalert box inside is shot, improperly wired (or all 3?).