I'm pretty sure those numbers are all wrong.
Bickell has a 4M cap hit. TT has a 1.344M cap hit. 5.344M.
Klein has a 2.9M cap hit. The difference is 2.444; that much is simple math and can't really be disputed.
Also, with Shaw's old deal off the books, as well as Fleischmann, Ladd, Weise, Rozsival, Ehrhoff all coming off, there is no way they only have ~4M right now. The numbers you're using are not accurate.
Where are you getting 1.344 for TT?
His AAV is $1,460,833, which includes his performance bonuses, which he may or may not earn. So his actual cap hit will most likely be less than that.
https://www.capfriendly.com/players/teuvo-teravainen1
http://www.generalfanager.com/players/538
His actual cap hit last year was $894,167 :
https://www.capfriendly.com/teams/blackhawks/cap_tracker
If his cap hit is similar this year, then they really aren't saving any money because TT has to be replaced, and his replacement will cost a similar amount.
So really the only savings is Bickell for Klein, which is a savings of 1.1 mil.
Klein for Bickell and TT gives them 6 forwards, 7 dmen and 2 goalies. If the cap goes up to 72.8 mil, they will have roughly 7.8 mil in cap space to sign Panarin and 6 other forwards. Less actually since they need to leave space for call ups.
That isn't enough. Panarin and Shaw will take up most of that by themselves.
If we traded them McIlrath instead, that would give them another 2 mil probably, depending on how much McI gets, though I don't see them giving us TT for McI.
We could retain 50% on Klein. Klein at 1.45 mil would work for Chicago, I think. Maybe we can squeeze Pokka out of them too.