I have attached an example script of the JMP 3D profiler with Mesh.

JMP has a 3D scatterplot and a Surface Profiler.

It is not too difficult with a little scripting to create a 9 plot view, or use Graph Builder: use 2 factors one for X and one for Y, making sure the one with the quadratic effect is included, then create if the other 2 factors can have discrete values like -1, 0 and 1, use one factor for X grouping and another for Y grouping then color by the modeled strength. Then instead of points use a contour.

Another possibility is to create "pictures" of the 3D plots the multiple levels of the remaining factors. and store then in a table. JMP 14 allows pictures to be used as the graphic icons, so you could create a grid of 3D views.

If your model has a secondary relationship, where the model can be reparameterized (for example maybe a combination of sealing and cooling temperature), you might be able to use it to reduce the dimensionality. However, from my experience, these are more difficult to visualize.