Kelsey Seybold (healthcare provider) charging up front in full instead of billing insurance

My wife is pregnant and we just started all the regular pregnancy care with Kelsey Seybold, our healthcare provider in Sugar Land. But I'm disappointed that they won't bill insurance first and I'm looking at upwards of $4000 out of pocket for 3 genetic tests, in addition to $900 that we paid for our first OB/GYN pregnancy visit.

While we have pretty good health insurance with BCBS and these should be covered, it is like watching paint dry having to wait for the claims process to play out and hoping it works out properly for us to get any sort of refund. Does anyone with Kelsey Seybold (or maybe another provider) have any luck getting them to run the claims with insurance before billing you? I might even settle for them charging me the insurance-negotiated rate because right now it seems like they're charging me up front in full. What can I do or ask them specifically? The whole process is so complicated. Thanks!