What is the conflict in "The Bass, the River, and Sheila Mant"?

At one level, the central conflict in W.D. Wetherell’s short story “The Bass, the River, and Sheila Mant” seems to be between the narrator’s longing for Sheila and his longing for the bass. However, I think Sheila and the bass—a species of fish—are actually stand-ins for two aspects of the narrator’s nature and his choice between the two is critical for the development of his self. If we look closely at the way Sheila is described, we get a clue into what really attracts the narrator to her: her beauty and poise, of course, but also her popularity and her family’s social status.

There was a summer in my life when the only creature that seemed lovelier to me than a largemouth bass was Sheila Mant. I was fourteen. The Mants had rented the cottage next to ours on the river; with their parties, their frantic games of softball, their constant comings and goings, they appeared to me denizens of a brilliant existence.

The fact that Sheila is courted by older boys and sportsmen, the entire “Dartmouth heavyweight crew,” adds to her appeal. Because Sheila is such a rare commodity, the narrator too covets her. In this, Sheila represents the part of the narrator’s nature that is easily seduced by the superficial and the glitzy. It is important to note that his crush on Sheila is not based on who Sheila is as a person.
On the other hand, fishing is a great passion for the narrator, an activity he is perfectly happy doing in solitude. He doesn’t need an audience for fishing; thus, like reading or hiking often is for other people, this is an activity arising from his deeper nature.

Automatically, without thinking about it, I mounted my Mitchell reel on my Pfleuger spinning rod and stuck it in the stern. I say automatically, because I never went anywhere that summer without a fishing rod.

From these lines we can see that the narrator almost feels incomplete without his fishing rod. Thus, fishing, the river, and the bass represent the more authentic, less self-conscious aspect of the narrator’s nature. The clash between the two aspects of his nature is comically highlighted when he actually manages to get Sheila—the object of his affection—out for a date in his canoe. While Sheila is talking to him, his attention is entirely on the bass he has accidentally pulled in and on trying to keep Sheila from discovering this. Earlier, Sheila has emphatically confessed she thinks fishing is “dumb,” which make sit even more urgent for him to conceal the hooked bass from her. But like the narrator’s subconscious, the bass makes its presence felt.

Downstream, an awesome distance downstream, it jumped clear of the water, landing with a concussion heavy enough to ripple the entire river. For a moment, I thought it was gone, but then the rod was bending again, the tip dancing into the water. Slowly, not making any motion that might alert Sheila, I reached down to tighten the drag.

The narrator’s frequent use of fishing words such as “drag” and “tugging” also embody the push-pull he feels between Sheila and the bass. Similarly, much as the “fish” is lured to the bait, the narrator too is lured by the glitter of all that Sheila represents. It is also interesting to see how the bits of conversation the narrator recalls with Sheila paint her as a self-absorbed and vain person. This could have little to do with Sheila herself and more to do with the little interest the narrator discovers in her company and his distracted state of mind.

I have to be careful with my complexion. I tan, but in segments. I can’t figure out if it’s even worth it. I wouldn’t even do it probably. I saw Jackie Kennedy in Boston, and she wasn’t tan at all.

Sheila’s empty words above are in contrast to how the narrator sees he physicality, “her lithe figure” and “the proud tilt of her shoulders.” Again, we see the narrator judge Sheila less by her personal qualities and more by her appearance.
When he finally cuts the line and lets the fish go, the effort it costs him is almost spiritual, as if he is amputating his soul. Although he doesn’t know it himself quite yet, the reader understand Sheila is already lost to him. We know the narrator’s enjoyment in the evening is ruined even before it begins; so much so that when Sheila decides to go home in Eric Caswell’s Corvette, all he can mumble is an indifferent okay.
Wetherell resolves the narrator’s psychological conflict with a neat lesson, with the narrator declaring that “he never made the same mistake again.” The mistake wasn’t about Sheila or the bass but about not listening to his heart and pretending to be someone other than he is. He now understands that he has to choose his deeper self over artificial glamour or the social pressure of seeming cool. Another lesson he learns is to value people by more than their appearance. Thus, as a young 14-year-old, he learns important, foundational lessons in self-hood.

W. D. Weatherell's short story "The Bass, the River, and Sheila Mant" is about a fourteen-year-old boy who develops a crush on a seventeen-year-old girl named Sheila Mant. 
Sheila is clearly out of his league. She is older, beautiful, and has captured the attention of an Ivy League rowing team. This causes an internal conflict on the part of the narrator. Should he ask her out? Here's how he describes his feelings about that conflict:

It was late August by the time I got up the nerve to ask her out. The tortured will-I's, won't I's, the agonized indecision over what to say, the false starts toward her house and the embarrassed retreats--the details of these have been seared from my memory.

Another internal conflict develops later as the narrator must choose between Sheila and the magnificent bass he secretly caught while on the date. He chooses Sheila, a decision he will soon regret.
You would expect to see an external conflict between Sheila and the narrator, since their date wasn't particularly successful. But Weatherell isn't as interested in the conflict between them as he is in the conflict that takes place inside the narrator. When Sheila goes home with another guy in his Corvette, the narrator doesn't object.
Weatherell resolves the internal conflict in the story's final paragraph, as the narrator comes to a personal realization:

Before the month was over, the spell she cast over me was gone, but the memory of that lost bass haunted me all summer and haunts me still. There would be other Sheila Mants in my life, other fish, and though I came close once or twice, it was these secret, hidden tuggings in the night that claimed me, and I never made the same mistake again.