Is Kilgore Trout a simple name-play on Theodore Sturgeon (if at all), or does a comparison go further?
Yep,
Trout is inspired by Sturgeon, but the stories are really a parody of the “sci-fi” genre as a whole.
Sturgeon also went through a number of hard times induced by vicious cases of writer’s block. It was well known in the field how other writers came to his rescue. The Trout character’s misery and poverty, despite his prolificness and imagination, can be read as another level of satire.
A few stories were published as by Kilgore Trout that were written by Philip Jose Farmer, until Vonnegut put a stop to them. This has been another source of confusion about the name.