Hi Giles,
I am afraid, current implementation of TabBarSpilitter control does not support this feature. However thanks for your nice suggestion. I have created a feature request # 1081 in this regard.
Feature Request #1081
You could track the status of the request in the above link. Please let me know if you have any questions.
Thanks for your interest in Syncfusion Products.
Best Regards,
Haneef