Are you restricted to using groups or LPEs? Could you use a compound path instead (Path menu > Combine)? Or is there some reason why an extension can't be used?
I'm afraid I'm not sure what you mean about a pincushion shape. I've seen pincushions in countless different shapes....but mostly they are roundish, or maybe something like a mushroom cap. Or rectangular - I've seen sort of wrist cuffs or bracelet with a pincushion attached to it, for seamstresses.
In much older versions of Inkscape, there is a functionality called "node sculpting" that could possibly be used for this. Recent versions of Inkscape also have node sculpting, but the recent versions they've changed the sculpt behavior from acting like a rubber band (each node moves a tiny but further than the last node) to acting like a bell curve (each node moves a tiny bit further from the last, up until a point, when each node starts moving a tiny bit less than the last). To me, the bell curve behavior is useless. I've asked to have it restored to the original behavior, or at least optional, but so far, I haven't seen it happen.
If you'd like to test, out of curiosity -- to use node scupting, use the Node tool. Select several nodes (on either a single path or a compound path), hold the Alt key, grab one node and drag. Then you can see the effect happening as you drag. And just stop dragging when you're happy with it. I don't know, maybe you'd like, or find some use for the bell curve type of effect?