I think you need an SSR for each element, with the HLT PID driving two SSRs, and the BK PID driving two SSRs. Of course, there may be more elegant ways to do it.
If I understand correctly:
You will have two 120v, 20a feeds, say F1 and F2. These will have to be on different circuits, and you could use 2000w elements with them.
You will have two elements in the HLT and two in the BK, say T1, T2, B1, B2.
You will have a PID for the HLT and a PID for the BK, say PT and PB.
You could use a 20a 3-position switch (or a lower rated switch with an SPDT contactor) to switch F1 between T1 - Off - B1. F1 would have to run to both T1 and B1 (each through an SSR). So you are at most powering 1 element with F1.
Similary, you would do the same with F2 to switch between T2 - Off - B2, so you are at most powering 1 element with F2.
Combining the two switches:
T1 T2 enables both elements in the HLT
T1 Off enables one element in the HLT
T1 B2 enables one element in the HLT and one in the BK
Off T2 enables one element in the HLT
Off Off ensures that no elements are enabled (great safety feature)
Off B2 enables on element in the BK
B1 T2 enables one element in the BK and one in the HLT
B1 Off enables one element in the BK
B1 B2 enables both elements in the BK
There is another way to approach it. If you don't really care to distinguish between, say T1 Off and Off T2, you could reduce this to four states:
T1 T2 enables both elements in the HLT
T2 B1 enables one element in the HLT and one in the BK
Off Off ensures that no elements are on (great safety feature)
B1 B2 enables both elements in the BK
For this, you could run both F1 and F2 through a simple 2-pole, normally open, on/off switch (with a contactor if needed).
After the on/off switch use a 3-position switch to switch between HLT / Split / BK, with three SPDT contactors that closed and opened the appropriate circuits. The load side of these contactors would run to the line sides of the appropriate SSR.