I'm not really sure how brick product works but I nearly understand powertopes (not sure how brick product relates to it.

With the 600-cell, it is the convex hull of 5 24-cells compounded together. remove the vertices of one and you get sadi (snub dis 24-cell) remove another (I don't think it matter which one

) you get Bidex (bi icosatetra dimished 600-cell) which has no tetrahedra but has 48 tridiminished icosahedra. (24 are from underneath the vertices that are removed from sadi, the others are diminishes if the 24 icosihedra in sadi. 3 5.g pyramid removed from each as well as removing the last 120 tetrahedra)

if it helps to study crown-jewels here is a description of the constructions for each of the "crown-jewels" (elementary Johnson solids)

Snub disphenoidthis is simply made by replacing the square faces of a square antiprism with a pair of triangles, the rest of the shape is stretched and squashed accordingly (to make it regular faced)

Snub square antiprismtruncate a square anti prism then alternate it or take a square and put a trangle at each edge than put two triangles between each "edge traingle (on the vertices) and take two of these structures and interlock them.

Sphenocoronatake an icosahedron and position it sao that so that it is standing on an edge and has a parallel edge on the the top, there are two opposite edges which are horizontal and in the middle of the shape and two which are vertical remove the pair of triangle at the vertical set and then cut what's left in half (you now have eight triangles (the result is open ended it is not closed up) this is the "corona"

put two squares that meet on the horizontal edge pair an then fill the empty space with 4 triangles (what you will have joined to "corona" is a "wedge-like structure made from two lunes [where a lune is two triangles either side of a square] this structure is call "spheno")

Augmented sphenocoronathis is just simply a sphenocorona with a square pyramid joined to one of the squares

Sphenomegacoronathis is "spheno" joined to "megacorona" the latter is the remainder of the icosahedron when "corona" is reemoved (or by putting those four triangles removed back on the "corona")

Hebesphenomegacorona above I described "megacorona", "hebespheno" is three lunes joined together (one is sandwiched between the other two) if you joined "spheno" and "hebespheno" you get an elongated 5-gon bipyramid

Disphenocingulum"cingulem" is a belt of 12 triangles similar to the 12 traingles in a 6-gon antiprism

join a "spheno" to either end so it's like a gryoelongation of two "spheno" stuctures

btw the structures above may need to be stretched and squashed before being joined together

Bilunabirotundatake a square and join a triangle to every edge of it then fold an opposite pair of traingles to nearly 90 degrees with the square. now take two such structures amd join the points of the two triange pairs, put a pentagon on either side of each pair of joined points (so there are two vertices with 3,5,3,5 configuration)

fold the pentagons (there are two above an two below assuming that the triangles joined are point are the horizontal and the squares are vertical) now fold the other triangles( the vertical set) so that they meet the penatgo pairs on th etop and bottom (so there are 4 vertices with a 3,5,5 configuration) doing this corectly all 4 vertices of both squares will be part of a 3,4,3,5 vertex configuration)

Triangular hebesphenorotundastart with a hexagon base and put a triangle on every other edge and put a "lune" (see above) placed on it's side on the other edges so that it alternates between lune and triangle, for eacj triangle (joined to the hexagon) fold it (and the lunes) so that each triangle and the two triangles from either side (for the lunes) go round the same vertex. put a pentagon at these intersection (of the three triangles round a vertex), put a traingle on the top of each square (part of the lunes) and fold them up, put a triangle on the top, the pentagons 5 edges will be joined to, the to triangles on two lunes (the bottom two) the two triangles placed on top of each lunen(the next two) and finally to the triangle at the top (the top edge) the point of the triangle place on top of the lunes will join to the vertices of the top triangle

I'm using these constructions to see if I can found analogies for 4D cases (now I'm sharing my secret) though I haven't found any "crown-jewels" yet