If it were mine ....
a. I would strip off the old finish ....
b. sand some of the rough spots a "little" bit ( 300 grit or so / and not too aggressively ) .. I'd also try some damp heat ( wet cloth and a steam iron ) to get some of the dents out as best you could...
c. I would not use any "filler" because I don't care what the can says ...it will never stain like the original wood ...!!
d. Depending on what might hide the scratches better - I would probably darken the wood a little with stain / and use a good varnish --- about 6 coats ...and probably a semi-gloss rather than a Satin ....
a product I like is -
http://www.brownells.com/.aspx/pid=5...UNSTOCK_FINISH
and I've used it on several Browning O/U's. The marketing label says "oil" - but its really a "varnish". It takes patience / 3 days per coat / let it harden - light sand in between - and then another coat. So it takes about 3 weeks just to do the finish.