Split Button or Menu Button?

I am a UI designer working on a project and the UX is asking for a Split Button button but the wires I received feel more like a menu button. I am wondering if a split button or even a menu button makes sense here. We want to introduce a split button on a existing button on the page. Today we have refer new client and Start New Loan, the proposed UX (Screenshot below) is to have Start New Loan now become a split button with the New Application and Share Application Link within it. The problem I am seeing is we are now going to have Start New loan now called New application. Should we not just have the share Application Link be the only option in the menu and let the Start New Loan CTA reamin the primary action of the split button or should we introduce a menu button.

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