"The Internet router is controlled only by our ISP and said unless we upgrade to their MPLS service they can't do anything!"

Normally what this means is:

"The Internet router is controlled only by our ISP and said unless we upgrade to their MPLS service they won't do anything!"

Unclear how using MPLS would be of much benefit since you did mention Internet.

That aside, on the 3750, I believe you can at least police traffic rates, so it may be possible to limit how fast the FTP server can push data to other hosts. Ideally, such a restriction only toward bandwidth constrained paths. Better yet would be a shaper, might not be possible on a normal 3750. (Either, as a solution is far from perfect since the FTP server can't just use excess available bandwidth, and whatever you limit it to, outbound, can at times, still be too much.)

Traffic being received by the FTP server is going to continue to be an issue. Reason being, controls downstream of the congestion point, usually aren't very effective unless very severe.

What you really want to control is the (QoS) queuing policy at the congestion points, for instance the two ends of your Internet connection. If your ISP is unwilling to take any action beyond using MPLS (or the other "popular" ISP solution, "you need more bandwidth"), you might shop for a more cooperative ISP (they're hard to find in this regard). Sometimes it's of benefit to mention to the current ISP you're doing so.

You're long term solution are either additional bandwidth and/or better bandwidth management. Besides whatever your local ISPs offer, you might also investigate whether there's a local Internet peering exchange.