I have two 12's. A 1968 EKO Ranger and a two year old Takamine. I Shubb's 12-string capo and have no problem as you describe. I used to get a great deal of sticking, detuning, twanging and vibration of various amounts and combinations (especially on the first fret) but the Shubb 'cured it'
The gentleman who re-set my EKO and reconditioned it told me never to de-tune my guitar. I know many do and I have heard their results. Good! He says there is no need to do it though, especially on newer guitars in concert pitch as the necks have been made to withstand anything these days. I have taken him at his word and the action on both guitars (the EKO the lowered) has remained constand with no 'bending up' of the neck. Sorry for non technical words but I do not know the full terms in some case.
I really was worried about the EKO not being detuned but, to date, he has been shown to be right. It could come down to preference I suppose.
As for Shubb 12 string capo... they play just as well on my six strings but never let me down on the 12's. I have 4 of them, often borrowed by other singers/band members on a night. No-one has ever complained.
This is based on my owm experience though so go with what you think is best. Lots of good advice on here :-)