Just inherited my Grandfather's Winchester 1894 in .30 WCF manufactured in 1897. The barrel is stamped "Nickel Steel Barrel Especially for Smokeless Powder".

What are the odds that a good gunsmith will tell me it is safe for limited shooting, perhaps after some unforeseen repair?

99%. unless it has some defect it will be ok.

If its in decent shape its as good and probably better than the mod. 94s built today and its certainly worth a lot more money than a new one. I have several that belonged to my granddad and my dad, and I've been shooting them for 70 plus years and have shot deer, bear and elk with them. your in good shape my friend.