I got in the Big Shuriken 3 and Phanteks PH-TC12LS! Complete weights for heatsinks, fans, and fin gap measurements below.
The BS3 has a small aluminum shroud that would block some airflow to the fins, which is held on by four screws. Interestingly, two of these screws also hold on three free-floating (non-soldered or connected) fins which isn't something I've encountered before. The BS3 has no gap between the mounting mechanism and the beginning of the fins, though there are three different areas of fin height, varying between 32mm, 23mm, and 14mm above the RAM.
Total weight: 474g
Heatsink only weight: 361g
120x120x17mm fan weight: 98g
Aluminum shroud: 15g
Compared to the BS3, the PH-TC12LS is pretty simple. There is a small shroud but it covers very little of the side of the fins, and no funny free-floating fins here. The fins are uniform in height, with 13mm of space between the mounting mechanism and fin stack, which is 19mm thick. The fins themselves are not totally open on the sides, only having 9mm of open space on the side of the fins that would allow air to pass through from the bottom or top (as mounted).
Total weight: 490g
Heatsink only weight: 300g
120x120x25mm fan weight: 160g
Aluminum shroud (and screws): 30g
I think the BS3 is going to blow the PH-TC12LS out of the water, but it's still going to be curious to see how wide the gap is.