Channel Balance Confirmation
I'm also not totally sure about the double spring. Does the other balance in the same window (marked 2740 EH B.S.I.) also have a double spring? Also, what is the weight of your sash? It may be possible that a regular balance would work for you.
As for the take out clip, it is necessary to install new balances. Then again, I would also say that its necessary to take the old ones out. How did you remove them without it? Check out the video below to see the proper method of replacing this type of channel balance.
The sash weight is approximately 18 lbs.
None of the windows in the house have takeout clips. When the sash stops are removed, the sash can be lifted beyond the point where there is any tension on the balance, hence not needing a takeout clip.

Great, thanks for the pictures. I'm not sure why one balance has a double spring, but a sash weight of 18 pounds isn't out of the ordinary.
Are you sure that 18 pounds accurate though? It's important to get a weight that's as exact as possible. The reason I ask is because two balances with a 2740 option can carry a window sash weight of 19 through 26 pounds, which would be too strong for your window. If your sash weight is definitely 18 pounds exactly, then I would go with the 2730 option instead. This can carry 13 through 19 pounds.
I would also double check the depth of your shoe. It looks a little bigger than the 15-002 that comes with our Series 351 balance. You might be looking at a Series 390 instead, which uses the 15-001 shoe. Check out the dimensions below to see which is closer.
I know it seems like splitting hairs, but it's important to be as precise as possible when determining the replacement balance you need.