I don't see the possibility of one trunk with hot amperage making all four ring.
I agree that is in the sucking category, but try another test that may sort things out, maybe not.
Parallel each line to both the 7100 and a single line set, preferable an old Bell/AT&T/GTE/Northcom type with a REN of 1.0.
When the 7100 goes nuts, do all four s/l phones ring or only one s/l and the four ports on the 7100?
One more test: With the lines unplugged from any equipment, put a tone generator on line one and using a buttset with the speaker on high, climb on line two, then line three, then line four. See if you hear a bleed through on any of them, if so there is not sufficient isolation from the Telco side.
Repeat this test with the tone on line two and the buttset on line three and then line four.
Repeat this test with the tone on line three and the buttset on line four.
If no bleed through on any combination, you have good isolation from the Telco and put that possibility to rest.
Does this craziness start only when line one rings? You could short out line one and see if the 7100 remains calm with line two, three or four ringing. That would sort it down to line one only.
One last test, reverse the order, put line four into line one position, line three in line two position. This may be meaningless if you have swapped the 4TRM but they then could have both come from the same mfg. batch.
Remember that when the guy said, "Cheer up, things could be worse." He cheered up and sure enough things got worse.
