While professional musicians need to have a separate policy for their instruments, you should be able to just add this to your homeowner's policy inventory list, and at no extra cost.  

But yes, you'll need to have a written appraisal on file in case there's a loss.  Appraisals are always subjective, to a certain extent.  If you take it to one luthier, point out the value added by your uncle, and get a figure you're not happy with, there's nothing stopping you from going to someone else.  Maybe he'll see things more your way.  

This really shouldn't concern you much, unless you live in a high-crime area, own large dogs and often leave the instrument lying around, or are prone to leaving it behind at coffee shops.  And if there is a loss, the hassle of processing a claim may not be worth the dollar value; insurance isn't really all that useful for instruments in the four-figure range.