The linked product is a mix of stearic and palmitic acid, with a higher SAP value than the stearic acid listed in most lye calculators. I've used similar products and it works fine in shaving soap, just make sure to use the correct SAP value (which for this product seems to be 207-211)
See...
I was thinking all mechanical, no melting. Quicker if one has the equipment.
The new recipe looks great, very close to one of my soaps (which I liked).
I have done a few formulations in the neighborhood of your intended soap, but with more CO and less lard. It performs well even for me with whiskers of Swedish steel :)
When it comes to combining the two parts I would make the new part separately and then use mechanical work to combine the two...
I would have swapped the SA/CO percentages (50% Stearic acid, 30% Coconut oil). That might work.
Usually shaving soap makers aim for Stearic + Palmitic >= 50
What was the temperature? I've had soap seize when soaping too hot. The fast reaction of KOH + SA then creates enough heat to boil off a lot of water.
I usually keep it at 75°C and 25-27% lye concentration to avoid this.