Well, the Little Scout is a very weak rolling block action, not a falling block, and has IMHO no value whatsoever except as an example of what kind of stuff was sold as a "boy's rifle" by Stevens after Page took over in 1906.  Even in good condition they are not strong enough for any modern .22 ammo except maybe CCI Quiet or CB Caps.  I honestly would not pay anything over $50 for one, even if it were well-nigh mint. 

I have no idea what a K02 would signify.  Maybe someone else can help?   

:::::  It came to me just now - the K02 stamped on the bottom tang could be the "serial number" of a Favorite.  They made so many that they couldn't use a simple ascending number sequence, so they started using  a letter/number combination, which nobody today can make sense of, since they apparently used some combinations more than once.  The 1894 patent date would make it sometime between 1894 and WW1. If it's in decent shape, $300 is a fair price.  If the bore is rusted or shot out, it's an easy re-line job, or you can simply fit a new barrel.  A Favorite is a desirable little rifle.  Really good ones can go for $600.    
  

Over the last several months, I have amused myself by watching Stevens Favorites' "serial numbers" and I use that term loosely.  Having one of these letter/numbers doesn't necessarily make it a Favorite but it suggests that it is.  Looking at any that I can, mostly Gunbroker ads, I have found two model 101's in 44 shot marked Q5xx and H6xx and a Marksman marked G9xx.  Whether these are serials or batch numbers, I will plead ignorance.  Note that Remington m1875 pistols were marked under the grips with batch serials explaining why sellers claim "first year of production" for them.

I have a very nice 14 1/2 22 long rifle stamped "regular cartridges" which I take to mean black powder or standard velocity, and marked SVG for Savage suggesting not intended for the new-fangled high velocity stuff.  If a CB cap doesn't make it out the muzzle, some old timers would try to "shoot it out" which caused ringed barrels or worse - not recommended.  I talked the seller down to $200 for this one.

James

Model 12 and 101 are also marked that way, but not easy to mistake for a falling-block.  They are, IMHO, the best of all the Stevens boys' rifles, including the Favorite.