tl;dr: It seems doubtful an antitrust claim would succeed.
Twitter would likely file a motion to dismiss under Fed. Rules. of Civ. Pro 12(b)(6) for failure to state a claim upon which relief can be granted. To your benefit, the judge would tend to construe all of your factual allegations as true (i.e. that Twitter excludes some websites but not others). Then, given this construction it would ask whether your claim is "plausible on its face," where plausibility means "pleaded factual content allows the court to draw the reasonable inference that the defendant is liable for the misconduct alleged." Bell Atlantic v. Twombly, 550 U.S. 544 (2007).
Note, in this second step, we're not only looking for factual sufficiency, but also legal sufficiency. The antitrust claim would likely falter for lack of legal sufficiency.
Your legal sufficiency would need to come from an alleged violation, so say you alleged an antitrust violation under Section 1 of the Sherman Act (15 U.S.C. §§ 1-7, amended at 12-27). In this case, you'd want to show 1) an agreement 2) that unreasonably restrains competition and 3) impacts interstate commerce.
If there isn't an agreement (e.g. Twitter just chooses to plug SoundCloud), that makes antitrust success even less likely, so imagine you were able to show evidence of an agreement. Then the trouble would be showing this is an unreasonable restraint of competition. Antitrust is a really complicated field, but the short version is: partnership and product placement agreements aren't necessarily violations of antitrust law---especially when the companies aren't in the same market. While there certainly differences, your question is similar (from a legal concepts perspective) to the one LetsListen informally made with respect to Facebook.
Aside: another issue is that a Twitter--SoundCloud collaboration isn't a within-market merger, which means the usual antitrust analysis in the Horizontal Merger Guidelines doesn't translate perfectly.