Kimbo,
Be wary of using one of those pumps in a blocked " pipe of the pan " situation , you might get a shitty back blast like my big bro did .
When my eldest girl with long hair ( now 43 ) & lived at home , we had a similar septic tank set up to yours .
On advice from Tony the owner of the sludge gulping company that emptied the tanks on request . I used to trickle two table spoons of caustic soda crystals down the plug hole. It didn't harm the action of the tank and it also helped to prevent long hairs being sucked into his sluge gulper tankers which sometimes caused him problems.
The bathroom sink & bath had those pop up plunger type plugs which always seemed to catch hair on the sink's flanged drain mounting plate a couple of inches down below the plug .
First table spoon of crystals was placed in the plug hole as the last bit of water was running out , the second was put down the wet plug hole then left it over night to dissolve hair and pipe crud .
I ensured I was in the bathroom first thing before the others in the house to wash it down the drains with very hot water in the morning .
I still do the same with our current bathroom as both Alison & our 13 yr old Munchkin have long hair , also adding in a " shallow shower " P" trap that runs out under a concrete heated floor shower drain to the list .
When you put the hot water down the plug holes use a kettle full for each hole and make the water in it not quite boiling , for caustic and really hot water are about the best hair and grease removers you can find.
Obviously do read the chemical hazard info on the container , wear rubber gloves a use eye protection etc. etc.