Are there third-party billing platforms where I can collect payments, take a cut, and forward the rest to merchants?

Various companies on the Internet will handle payments/receivables for you. You can Google for them (payroll services, invoice outsourcing, payments outsourcing, etc.) , their are literally 100s of them. For a fee, they will bill the party and pay you back. If you in turn have a bill to pay, they can pay that for you as well.

Most of them have an online system where such receivables and payments instructions can be setup. Some of the advance companies have intelligent system to take (receive) a payment and then do further processing on it - as per your instructions (which is what you are seeking).

Some companies: www.nationalach.com, www.achdirect.com, www.paysimple.com, www.nationalprocessing.com, www.aramorpayments.com