Ask HN: Top Payroll SaaS with API?

My company is a service company, but we've built a fairly robust job management system to manage all of our work, including tracking pay for employees and subcontractors. Right now, our payroll provider works fine, but I would like to look into finding one that offers an API to manage payroll. All of our pay data is already located in our own job management application. Ideally, I would like to be able to click a button that says "Run payroll for XX week", and our app send all of the pay data to the payroll provider auto-magically. Even just manually copying data for 100 employees every two weeks is quite time consuming. I did a little searching, only to come up with Gusto/ZenPayroll, but they appear to right now be limiting their API to large partners that will expand the Gusto audience. Are there other options on the market I'm not finding?