I realized that I can use the section function in Stella to systematically go through all convex uniform polychora and identify the possible cupolas/rotundas/pyramids, as well as diminishings.
Of course, most of them will be known...
octahedral pyramid (half of hexadecachoron)
octahedral cupola (half of icositetrachoron)
cubic pyramid (vertex cut from icositetrachoron, changing neighbouring octahedra into square pyramids)
icosahedral pyramid (vertex cut from 600-cell)
octahedron||truncated octahedron (half of truncated hexadecachoron, equatorial octahedra cut into square pyramids)
icosahedral rotunda (icosahedron cut from truncated 600-cell, not a segmentochoron: 1 icosahedron, 20 truncated tetrahedra, 12 pentagonal pyramids and 1 truncated icosahedron)
triangular prismatic pyramid and triangular wedge (rectified pentachoron can be cut into these two segmentochora)
tetrahedral cupola (tetrahedron cut off rectified tesseract, if two opposite tetrahedra are cut, segmentochoron 1-1-0||0-1-1 results)
cuboctahedral cupola (cuboctahedron cut off rectified icositetrachoron)
icosidodecahedral rotunda (icosidodecahedron cut off rectified 120-cell)
For rectified 600-cell, three shapes result by cutting off an icosahedral cell. One layer results in icosahedral cupola (1 icosahedron, 20 octahedra, 12 pentagonal pyramids, 1 icosidodecahedron), second layer results in icosidodecahedral copula (1 icosidodecahedron, 20 octahedra, 12 pentagonal antiprisms, 30 square pyramids, 1 rhombicosidodecahedron), but both layers can be taken together into extended icosahedral cupola (1 icosahedron, 40 octahedra, 12 gyroelongated pentagonal pyramids, 30 square pyramids, 1 rhombicosidodecahedron)
pentagonal prismatic pyramid (vertex cut off rectified 600-cell
octahedron||truncated tetrahedron (octahedron cut off small rhombated pentachoron, the rest of the polychoron forms segmentochoron truncated tetrahedron||cuboctahedron
triangle||hexagonal prism (triangular face (cuboctahedron/cuboctahedron) cut off small rhombated pentachoron)
rhombicuboctahedron||truncated cube (rhombicuboctahedron cut off small rhombated tesseract)
square||octagonal prism (square face (rhombicuboctahedron/rhombicuboctahedron) cut off small rhombated tesseract)
cuboctahedron||truncated cube (cuboctahedron cut off small rhombated icositetrachoron)
cutting off second layer of vertices results in truncated cube||great cuboctahedron
cutting both at once will result in a shape consisting of 1 cuboctahedron, 6 elongated square cupolas, 20 triangular prisms (8 + 12), 8 triangular cupolas and 1 great cuboctahedron
rhombicosidodecahedral cupola (rhombicosidodecahedron cut off small rhombated 120-cell)
pentagon||decagonal prism (pentagon cut off small rhombated 120-cell)
icosidodecahedron||truncated icosahedron (icosidodecahedron cut off small rhombated 600-cell)
rhombicosidodecahedral rotunda formed by next layer of small rhombated 600-cell
runcinated icosidodecahedral rotunda, formed by connecting of previous two
truncated tetrahedron||truncated octahedron (truncated tetrahedron cut off prismatorhombated pentachoron)
the rest of the body gives a shape made of 1 cuboctahedron, 4 truncated tetrahedra, 6 hexagonal prisms, 4 triangular prisms, 4 triangular cupolas and 1 truncated cuboctahedron. I suspect it might be possible to glue these two parts of prismatotruncated pentachoron together the "wrong way", so that the the triangular cupolas will be joined to hexagonal prisms and truncated tetrahedra to triangular cupolas instead of combining the cupolas to form cuboctahedra. Not sure if the joins won't blend (into augmented truncated tetrahedra and/or elongated triangular cupolas), but either way, it should be a possible shape!
Prismatotruncated tesseract has an "equatorial region" formed by great cuboctahedral prism and two caps, each of them being formed by 1 rhombicuboctahedron, 6 cubes, 12 hexagonal prisms, 8 truncated tetrahedra, 6 square cupolas and 1 great cuboctahedron.
Prismatotruncated hexadecachoron has caps as truncated cube||great cuboctahedron
Prismatotruncated icositetrachoron has caps as truncated octahedron||great cuboctahedron
Prismatotruncated 120-cell has nonsegmentochoric caps which take 3 layers of vertices. They consist of 1 rhombicosidodecahedron, 12 pentagonal prisms, 20 truncated tetrahedra, 30 hexagonal prisms, 12 pentagonal cupolas and 1 great icosidodecahedron.
Prismatotruncated 600-cell has segmentochoric caps as truncated dodecahedron||great icosidodecahedron.
Small prismated decachoron can be cut in two halves which are tetrahedron||cuboctahedron. It might be possible to glue them together the wrong way.
Small prismated tesseract can be cut in two cube||rhombicuboctahedron and an equatorial rhombicuboctahedral prism.
Small prismated 120-cell has caps formed by dodecahedron||rhombicosidodecahedron.
Small prismated icositetrachoron has caps formed by octahedron||cuboctahedron.
Sorry for hard-to-read list, I was writing as I was discovering
Note that some small rhombated polychora have two different ways of diminishing. Small rhombated 120-cell, in particular, can have either rhombicosidodecahedra removed, which removes 30 triangular prisms and 20 octahedra, diminishes 12 rhombicosidodecahedra around it, and adds 1 great icosidodecahedron, or it can have a pentagon removed, which removes five triangular prisms, cuts in half 5 octahedra, diminishes two rhombicosidodecahedra and adds a decagonal prism. And both types of diminishing can be used at once! That will be a lot of shapes.