Most pics I can find of the Hot Rock show a standard 1 bolt adjustable seatpost, or the older style pinch bolt. I wonder if the previous owner swapped something else in…

Anyway, best seat for the groms that I’ve found is the SDG Fly Jr. It’s really fantastic. Comfy (10 yr old very petite daughter and normal sized 7yr old son never complain of sore buts) super durable, and comes in sweet colors. Oh, and it’s cheap ($35).

 

I'm not much help on that seat. I think what you are describing is a bmx specific "seat" that isn't meant to be sat on much. Is the post plastic? Seat angle adjustable? I'm not sure there is a standard for those since the seat and post often come as a pair: Seat & Post Combo Seats Freestyle Parts

For a traditional mount kids seat, we have the BW and the SDG Junior. My daughter likes both but when she was younger she preferred the BW, which is also half the cost:

Amazon.com : BW Bicycles Kids Seat – Youth Replacement Bike Saddle – Bue : Sports & Outdoors

Amazon.com : BW Bicycles Kids Seat – Youth Replacement Bike Saddle – Bue : Sports & Outdoors

smile.amazon.com