Alright, to start off, I have some questions for you.
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Are these the only champions you have that you own some, but not all, of their skins? (Karma, Sona, Twisted Fate, Zyra, Ezreal)
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If so, how many would you be removing?
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Are there any with only a few unowned skins remaining?
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Have you checked whether the skins will still be available to your account even if the champion is not, or will you have to repurchase them?
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Are there any champions you own all of their available skins?
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Assuming removing a champion from your account also removes any skins owned for that champion, do you have any Victorious or Special Edition skins only available through special means (Rusty Blitzcrank, King Rammus, etc..)?
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If you answered yes to either question, don't remove them from your account. The former won't count towards the 10 unowned Skin requirement, and they'll help fill out your champion roster. The latter, assuming they're removed, will likely not be recoverable.
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Do you own any recently released (expensive) champions, or any with very few skins available? (All champions below cost 6300 IP unless listed otherwise)
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1 skin: Illaoi, Jhin, Kindred, Tahm Kench
Total: 4 champions for the roster, 4 total skins, 25,200 IP
- 2 skins: Azir, Bard, Diana, Ekko, Elise (Mystery Gift doesn't give Victorious Skins, I do believe), Kalista, Kha'zix (until Death Blossom's released), Lissandra, Rek'Sai, Vel'koz, Yasuo, Yorick (4,800 IP), Zac
Total: 13 champions, 26 total skins, 80,400 IP
- 3 skins: Aatrox, Braum, Gnar, Jayce, Lucian, Nami, Quinn, Rengar, Rumble (4,800 IP), Swain (4,800 IP), Taric (1,350 IP), Urgot (3,150 IP), Viktor (4,800 IP), Zed
Total: 14 champions, 45 total skins, 75,600 IP
Okay, so...
Best case scenario, you get Traditional Karma, but have to purchase all the champions you're wanting/needing again.
However, let's say you're down to 10 unowned skins, receive the Mystery Gift, but it's not Traditional Karma.
That'll leave you at 9 unowned skins, and you'll be required to purchase another champion.
Most of the cheaper champions tend to have 4+ skins, while the majority of champions with 1-3 skins cost 6300 IP (with a few exceptions).
Assuming Traditional Karma take multiple attempts, I would suggest keeping a few extra skin slots/champions as backup, preferably any champions with very few skins available that you play/want their skins.
I'd say holding on to something like 14 unowned skins would be a decent number without heavily increasing the potential skin pool.
If nothing else, it'll help reduce the IP you'll need to purchase more champions while also filling up your champion roster.
If you still end up having issues getting the Tradition Karma, there are even a few cheaper champions that will only raise the skin pool by 2-3 skins.
Edit: Turns out there were more champions with 2 skins than I thought, and added those with 3 skins, some of which cost less IP (especially Urgot at 3,150 and Taric at 1,350). Also removed Zyra, forgot you're keeping her.