It is not possible to obtain all 3 of the new First Strike icons without spending a bit of RP (beyond what's needed to buy the skins in the first place). You don't get enough cores from the IP caches for that; altogether those 4 caches (and the single Augment Cache you can make with the fragments you get in the process) get you 12 guaranteed Project Cores with 2 fragments left over.
Best-case scenario: you can obtain enough Project Cores by buying, at minimum, 3 additional Augment Caches with 885 RP.
Worse-case scenario: you get unlucky and none of the Augment Caches you open contain a fragment for a new cache, so you need to buy 5 additional Augment Caches with 1475 RP.
In either case you need to open 5 Augment Caches. The best-case assumes that you get 4 more fragments during this; 1 to go with the 2 fragments you're guaranteed to have left over, and 3 more to make yet another cache.
Double-checking to make sure I thought it through correctly. I think I got it right.
https://i.imgur.com/1HvNiEO.png?noredirect