L2C,
I'm not sure you are aware of just how much effort would be required to replace your P trap (which is probably just fine). I had to replace the pump on our washer and that required removing the washer and then replacing it. Changing the P trap would require a similar amount of effort since you cannot access the plumbing behind the washer without removing it.
Just so you know, removing the washer (at least in my case) is no trivial task. You have to remove the trim piece as well as a retainer piece below the washer. (This requires cutting any caulk that may exist between your flooring and the trim piece.) You have to unhook the water and electrical, then you have to lift it over your flooring and slide it out. You need to purchase something to protect your flooring against damage when you move it around on the floor. This is all a lot of work unless it is absolutely necessary.
Have you tried just running the washer to see if some bleach will solve your problem?
Oh, and do NOT silicone around the drain hose from your washer to close the gap. You don't want this to be a closed system.
Bob