You have a Tuono? Is that not working out as the occasional track tool for you?
Are you trying to keep only one bike in the stable?
I ask, in that I would state that if you do own a Tuono, it is probably a better track machine out of the box, if not equally versatile. I cannot comment on the 2-up qualities of either the Tuono or 1290, but I can't envision a pillion being very happy for long, on either of these bike's perches.
I do not know what you mean by "fastest road group" ... Are we to assume that it is not the "race" or "expert" group at track days - ?
Assuming for the moment that what you're referring to is the "race" or "expert" group, I can't advocate enough having a dedicated track/race bike. Differences in setup alone, are too much effort to switch one's only bike back and forth for presumably frequent track days.
Assuming instead, that you mean the "Intermediate" level at track days, then I would first ensure that on any bike, the suspension is properly setup (professionally) with correct springs and valving for your use of the bike. The hard part is, having a compliant street bike setup where you can firm things appropriately for the occasional track day, AND at the same time enough damping and preload adjustment in the suspension for brisk 2-up riding.
Beyond suspension, I'd personally pop for a decent steering damper, and of course tyres suitable to skill level and need. Regardless the tyre manufacturer/model you like, the newer the tyres the better IMO. Ergonomics are important too, for both comfort and to keep from dragging the hard parts. Seat, bars, pegs/rearsets can all be addressed via the aftermarket.
Both a 1290 and Tuono have more power than anyone would ever require on the street (assuming the pillion doesn't weight 200 kg), so there's really nothing to do there. And both these bikes have enough juice out-of-the-box to satisfy even an expert level trackday junkie.
Ensuring that bike is in excellent operating condition - the track is no place to compromise.
Last, but most important is you. Preparation, fitness, appropriate mental attitude and focus are the keys to best track day experiences, regardless one's experience, abilities, or bike they are on.