I tried the 156 grain Wildcats in my 25/06AI (24 in 10 twist barrel) to see what velocity that case would produce with that bullet weight. It achieved 2,830 fps (63.0/H870), and as expected I got full bullet profile keyholes on the 50 yard target.
My 25/06AI was a standard 25/06 for the first 130 shots before it was rechambered to the AI, which produced an increase in velocity of about 225 fps. As an example with the 100 grain NBT, 62.0/Re 22 produced 3570 fps in 25/06AI, and 57.0/Re22 produced 3,345 fps in 25/06 mode.
Consequently, I estimate that the 25/06 would achieve about 2,600 fps with the 156 grain WC, and at that starting velocity, you certainly would not have a flat trajectory.
IMHO Re 22 would not be a suitable powder use with that bullet weight, as the starting load would be about 42.0 grains, and the maximum about 46.0 grains. This means there is a lot of air space in the case, even with the base of the 156 grain WC projecting about .375 below the neck/shoulder junction.
I suggest you take Kirby's excellent advice and not bother with trying the 156 grain WC's in your 25/06, as I believe it would not be worth the time, effort, and expense. Hope the above is of some help. Brian.