A while ago I was using .22 caliber nailgun charges in adapters to shoot .32 rimfire in a 1915 Favorite.  The lowest (coded grey if I remember right) were soft enough. The next up are brown, and they were also OK.  Noting that a 1915 is considerably stronger than an 1894, due to its' wider link. 

I tried a few brown coded .27 nailgun charges in a .25 Stevens 44, with no bullet, and had a dickens of a time getting them to extract.  Haven't continued with that experiment.   The 44 has been converted to centerfire, (by swapping in a CF breechblock),  and I'm using cases formed from Hornet brass.

It's possible, but laborious, to make passable .25 Stevens out of .17WSM, but a Favorite won't fire  them - the rims are too hard/thick.  I'm shooting those only in a 1938 Model 83 bolt action. 

If you can swallow the prices, there  is NOS ammo available on auction sites, and you can buy a kit from Sharpshooter Supply to reprime spent cases.   

I’ve breech seated and fired both the level 3 (green) and level 4 (yellow) .27 caliber nail gun loads in my 44 1/2 .25 Stevens, the green loads seemed really mild, the yellow loads had quite a bit more zing. The green loads extracted okay, but the yellow loads cases bulged and had to be removed with a cleaning rod. I found both loads to be very dirty to shoot with lots of fouling. It was also very easy to slip the nail gun loads past the extractor being such a short case, which was a pain, and difficult to remove. I much prefer shooting and using .17 wsm ammo converted to .25 Stevens in my rifles. Now I have only fired a few of the converted rounds made from .17 wsm in both a 1915 and 1894 favorite, but I think they all fired fine. 

I think your favorite would probably handle the .27 caliber brown loads (level 2) just fine.

No I didn’t check the velocity.
I used 55 grain cast bullets sized .251, meant for .25 acp, from Rim Rock bullets.

Scroll up from post #53 to see how he's made an adapter from .22 Hornet.  Lot of work, but once done it's reloadable.

Both my sideplate .25 Stevens Favorite (1892/93), which is in pretty good condition, and the Model 83 (1938) are .257.  Ditto my '94 Favorite, (post-1900), and both of my Model 12s (ca. 1912).   Where a .250 barrel fits in I haven't a clue. If they exist at all it must have been for a very short while.

Don't leave that 044-1/2 where I can see it.  It might grow legs!

uscra112
Your 25 caliber rifle bores of 257" groove diameter sure span a wide range of Steven's barrel production.
The attached photo shows a .257" lead bullet, that was driven through the barrel, of my Favorite, from the breech end.
Otto

Roller is stronger than  '94 Favorite, IMHO.   1915 Favorite maybe equal.  Difference is the width of the link.  The link takes quite lot of stress, since the rimfires were rarely fitted up so that the breechblock bears against the shoulders in the frame.