Here's an approach that could resolve your problem, and if not would help with troubleshooting.
Create a second Apache virtual server identical to the current one
Send all "normal" user traffic to the original virtual server
Send special or long-running traffic to the new virtual server
Special or long-running traffic could be report-generation, maintenance ops or anything else you don't expect to complete in <<1 second. This can happen serving APIs, not just web pages.
If your resource utilization is low but you still exceed MaxClients, the most likely answer is you have new connections arriving faster than they can be serviced. Putting any slow operations on a second virtual server will help prove if this is the case. Use the Apache access logs to quantify the effect.