First, construction lumber is a terrible choice for stair treads. The wood is much too soft and open grained to last under any traffic, even light stocking feet. The wood will splinter eventually, even under paint. Almost anything dropped on the wood will dent it and cause splinters feom the dent.

Now, to answer your question. Poly is almost impossible to spot repair. Any finish repair will mean recoating the entire tread. That is not really difficult on stair treads, however. Any finish will be absorbed into the wood so much, getting a working coating will be difficult. Reason, the wood is so soft, the fibers will swell and stick up. You would need to sand snooth. The next coat will also be absorbed about half as much. Several coats/sandings later you may have a workable coating. W#hich will quickly be degraded by the dogs feet.

I'd recommend the carpet now. Will save a potential splinter in someone's foot---including the dogs.