Try a 170 grain jacketed bullet and a smokeless powder load down under 30,000 psi.  If it shoots okay, then you know it's something with your bullet/load, and not the rifle.

The first thing I'd try with the lead bullet is something slower burning and more case filling than the Unique.  I'd think Varget or IMR 4895 in the 18 to 20 grain range would give you consistency without excessive pressure and would keep the velocity down.  You might have to go to a heavier bullet to get the velocity down to cast bullet range without using pistol powders. If you can get a gas check bullet, I'd try that. 

dave

Can we get a bit more background data
Twist rate
Bullet length
What distances have you tested

If you read Phil Sharpe's "Complete Guide to Handloading" in some detail, you will find references to his 1902 rolling block in 7x57. It was new when he got it.  In the late 30s it had a big Unertl scope.  He used it for testing European ammo and bullets.   

The book also has lots of loads for 7mm rollers.

Sharpe's book is now in the public domain and can be downloaded either free or for very little.

Found my old data card for my RB carbine. With the 169 grain RCBS bullet it liked 20 grains of 4198 or 16 grains of 2400 or 24 grains of 4895. With a 150 grain Lyman Loverin it liked 14 grains of 2400 or 20 grains of 4198 or especially 19 grains of 2015. I should add that the brass was concentrically fire formed to the chamber and thereafter only neck sized.

I think this one is fairly easy. I found a nice drawing of your bullet and ran the numbers through a stability program.

The stability of the 150gr bullet is quite high at 3.1 to 3.2 at MV. This is certainly high enough to lock the attitude of the bullet for a good distance from the muzzle.

All bullets kick out at the base as they leave the crown. Perfect crown with a perfect base kicks out the least. But if either of the two factors is less than perfect the amount the bullet kicks out increases. 

With a SF as high as 3.1 the bullet will be locked into this kicked out base position until drag can take effect and gradually reduce the tail wobble to stable flight. Your target distance of 25yds is likely showing you a snapshot of the bullet still frozen.

If you shoot a set at 100yds you would likely see a different picture. However if the bullet is wildly veering at the muzzle group size will suffer.

I shot a 45-70 at the Wilton match that didnt settle down until the 200 yd mark. The group size was significantly better at 200 than at 100. This was seen by many.

I've been using the RCBS  168 grain gas check bullet cast of #2 alloy for many years.  Sized to .285 and driven by 21.5 grains of 4198; CCI #200 primer.  Brass is made from military 30-06 which is a good bit heavier than commercial 7 mm brass.  This is a fun load, not a target load and works well in both my Remington rolling block rifle and carbine.   As I recall, the velocity out of the rifle is about 1825 fps.  The load was designed for the rifle, but works fine in the carbine too (luck).  According to Barnes, the original 7mm round used a 173 grain bullet at about 2296 fps.  That tells me that a heavier bullet should shoot better.
Of course that also assumes that your bore is half way decent.

Good luck,
George
Salmon, Idaho