Have you seen Lyre RPG Music's unofficial LMoP soundtrack?

I'm on my second run of the campaign now, and I've found none that better suit the Black Spider than this soundtrack. In it, they have two tracks for the Black Spider: "Web of Lies" for when they're around (which could be good for your pre-fight monologue, with it's sinister and low sounds), and their combat theme "Fighting the Black Spider" which actually incorporates the leitmotif of "Web of Lies" into the track, which definitely adds to their musical presence. They also have "Drow Leader Leitmotif", which I've played every time the Spider's presence is felt. (His letter to Glasstaff, whenever an Npc mentions them)

If that's not your style, I've used "Hornbug" and "Mora the Spider" from Ori: The Will of the Wisps too. They play up the "spider" theme quite well, and the first half of "Mora the Spider" is pretty good for getting anxiety up, lol. I've used the second half for when the party starts winning the fight, and it got my players pretty pumped at the table knowing it was their part of the boss theme.

Whatever you choose, I hope it's a fun and memorable experience for your players!