there is no way to get around the breeding limit unless you get lucky with twins on the last breeding. (which is pretty much impossible as twins are never a guarantee).
Chincotegues, Marwaris and Akhal-tekes are all limited to 5 live foals and 3 lab pulls. note that you can spend more money/ingots on the lab pulls for a chance of more eggs and straws, but I haven't found it worth the cost to do so though. It's easier to just buy new breeding stock off someone 

Thank you