No problem dirt biking up to the top of Railroad. The road is open and clear.
The basin should have some snow in it still, but most of it should be gone by now. Should be just enough to keep your fish cold. Though, be careful with bears. Last couple of times I have been up in that section of the woods (I often use this route to access the Big Boulder Chain Lakes after fishing Tin Cup/Gunsight), I have seen bear and quite a bit of sign. Most of the sighting and the sign is down lower in the forested basin between Tin Cup and Walker lakes; however, to play it safe you might want to hang your food if you are going to be gone from camp for several hours.
Which peaks are you looking at bagging? There are some pretty tough ones there (WCP-10 and the highpoint of the China Wall).